#6848254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6755450);