#396471

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

1430 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)(9,10,13,15)(11,16,12,14)(17,18,23,26,22,28,21,27)(19,24,31,32,20,30,29,25);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,20)(18,24,28,30)(21,29,23,31)(25,26,32,27);(2,4)(6,8)(17,20,22,19)(18,25,28,32)(21,29,23,31)(24,26,30,27);(9,12,13,11)(10,14,15,16)(18,26)(21,23)(24,32)(25,30)(27,28)(29,31)

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 303667);