#6770984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#38034	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6678180);