#6492959

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#29675	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6400155);