#512679

order	512
nilpotency class	5
isoclinism	isoclinic class of #512666
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	#3499	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 785 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419875);