#6814837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60770	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,20)(15,21)(16,22)(18,31)(23,53)(24,40)(25,41)(26,43)(28,39)(29,46)(30,60)(32,47)(33,48)(34,49)(35,65)(36,51)(37,52)(38,42)(44,56)(45,63)(50,62)(54,67)(55,72)(57,71)(58,69)(59,70)(61,66)(64,68);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,28)(13,24)(14,34)(15,38)(16,26)(18,44)(19,40)(20,49)(21,42)(22,43)(23,29)(25,36)(27,39)(30,57)(31,56)(32,54)(33,59)(35,55)(37,58)(41,51)(45,68)(46,53)(47,67)(48,70)(50,66)(52,69)(60,71)(61,62)(63,64)(65,72);(9,12,16,32)(10,18)(11,23)(13,31,15,29)(14,35,37,30)(17,39,43,67)(19,47)(20,50)(21,27)(22,46)(24,28)(25,57)(26,56)(33,62,36,63)(34,64)(38,54)(40,44,42,53)(41,68,70,66)(45,52)(48,60)(49,72,69,71)(51,65)(55,59)(58,61);(2,6)(5,7)(9,13,16,15)(10,19,22,21)(11,24,26,38)(12,29,32,31)(14,36,37,33)(17,40,43,42)(18,27,46,47)(20,51,52,48)(23,54,56,28)(25,58,59,34)(30,63,35,62)(39,53,67,44)(41,69,70,49)(45,65,50,60)(55,61,57,64)(66,71,68,72);(3,8)(5,7)(9,14,15,36,16,37,13,33)(10,20,21,51,22,52,19,48)(11,25,24,34,26,59,38,58)(12,30,29,62,32,35,31,63)(17,41,40,49,43,70,42,69)(18,45,27,60,46,50,47,65)(23,55,28,61,56,57,54,64)(39,66,44,71,67,68,53,72)

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722033);