#6367486

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1644 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)(3,8)(4,14)(5,11)(6,10)(7,12)(9,15)(13,16)(17,18,24,29)(19,30,25,32)(20,31,21,26)(22,28,23,27);(1,3,7,4)(2,8,12,14)(5,15,6,13)(9,10,16,11)(17,19,24,25)(18,20,29,21)(22,31,23,26)(27,32,28,30);(1,4,7,3)(2,9,12,16)(5,15,6,13)(8,10,14,11)(17,20,24,21)(18,25,29,19)(22,30,23,32)(26,27,31,28);(17,21,24,20)(18,26,29,31)(19,28,25,27)(22,30,23,32);(1,5,7,6)(2,10,12,11)(3,13,4,15)(8,9,14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274682);