#345623

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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

540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252819);