#512677

order	512
nilpotency class	5
isoclinism	isoclinic class of #512663
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 795 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=hi,f^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419873);