#337724

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

651 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)(17,18,21,25)(19,26,22,23)(20,28,30,31)(24,27,32,29);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,28,32,31);(1,4,5,8)(2,6,3,7)(10,13)(11,16)(14,15)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(2,3)(6,7)(17,20,21,30)(18,24,25,32)(19,27,22,29)(23,31,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 244920);