#396142

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=i,c^2=i,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 303338);