#403562

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 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)(3,4)(6,7)(17,18,21,26)(19,23,27,31)(20,24,22,25)(28,32,29,30);(5,6,8,7)(9,10,12,14)(11,15,16,13)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,30)(26,31);(2,4)(9,11,12,16)(10,13,14,15)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(5,7,8,6)(18,25)(20,22)(23,30)(24,26)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=gi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310758);