#6627697

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(9,10,14,13)(11,15,12,16)(17,18,22,26)(19,23,21,24)(20,25,28,32)(27,30,29,31);(2,5)(6,8)(9,11,14,12)(10,15,13,16)(17,19,22,21)(18,23,26,24)(20,27,28,29)(25,30,32,31);(9,10,14,13)(11,16,12,15)(18,24)(19,21)(20,28)(23,26)(25,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,13,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32);(9,13,14,10)(11,15,12,16)(20,28)(25,32)(27,29)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,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, 6534893);