#6367154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1552 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:

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

Polycyclic presentation:

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

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, 6274350);