#6367234

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

1604 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)(3,12)(4,9)(5,10)(6,13)(7,11)(8,15)(14,16)(17,18,21,24)(19,22,26,31)(20,23,28,25)(27,29,32,30);(1,3,7,14)(2,8,11,5)(4,15,6,10)(9,16,13,12)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,4,7,6)(2,9,11,13)(3,10,14,15)(5,16,8,12)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,5)(2,10)(3,13)(4,16)(6,12)(7,8)(9,14)(11,15);(2,11)(3,14)(9,13)(10,15)(18,24)(19,26)(23,25)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,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, 6274430);