#6614514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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)(3,8)(4,7)(5,6)(17,18,21,24,22,25,30,32)(19,26,27,28,23,31,29,20);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20)(18,23)(19,25)(21,26)(22,28)(24,29)(27,32)(30,31);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,26)(22,28)(23,25)(24,27)(29,32)(30,31);(10,13)(11,15)(19,23)(20,28)(26,31)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6521710);