#6770673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#38908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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:

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

Polycyclic presentation:

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

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, 6677869);