#338812

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

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

523 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:

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

Polycyclic presentation:

2^9|a^2=ei,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, 246008);