#315133

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,d^c=dh,f^c=fi,g^c=gi

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, 222329);