#7104812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7012008);