#6657873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,35)(37,39)(38,40);(2,5)(3,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,30,29)(26,31,34,33)(28,35,37,38)(32,39,40,36);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(2,5)(6,8)(10,17)(11,20)(16,22)(19,21)(25,27,30,29)(26,31,34,33)(28,35,37,38)(32,39,40,36);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,27,30,29)(26,33,34,31)(28,35,37,38)(32,36,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282682, #72283268, #72283966
maximal normal subgroup of	#72282682, #72283268, #72283966

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6565069);