#512665

order	512
nilpotency class	5
isoclinism	isoclinic class of #512663
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14,17,8)(4,9)(5,10)(6,12,27,11)(7,13)(15,40,43,28)(16,41,44,29)(18,30,26,31)(19,45,47,55)(20,32)(21,34)(22,33,50,39)(23,51,52,35)(24,37,53,36)(25,38)(42,60,61,54)(46,62,63,56)(48,58)(49,57,64,59);(1,3,7,18)(2,8,13,31)(4,19,21,15)(5,23,25,16)(6,26,27,17)(9,28,34,55)(10,35,38,29)(11,30,12,14)(20,42,48,46)(22,43,50,47)(24,44,53,52)(32,56,58,54)(33,40,39,45)(36,41,37,51)(49,63,64,61)(57,62,59,60);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,28)(10,32)(11,39)(12,33)(13,34)(14,40)(16,42)(17,43)(18,19)(23,46)(24,49)(25,48)(26,47)(27,50)(29,54)(30,45)(31,55)(35,56)(36,59)(37,57)(38,58)(41,60)(44,61)(51,62)(52,63)(53,64);(1,5,7,25)(2,10,13,38)(3,16,18,23)(4,20,21,48)(6,24,27,53)(8,29,31,35)(9,32,34,58)(11,36,12,37)(14,41,30,51)(15,42,19,46)(17,44,26,52)(22,49,50,64)(28,54,55,56)(33,57,39,59)(40,60,45,62)(43,61,47,63)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ehi,f^b=fgh,g^b=gi,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284440, #72287238, #72318941
maximal normal subgroup of	#72284440, #72287238, #72318941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419861);