#393568

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1098 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,19,21,24,29,26)(20,23,28,25,27,30,32,31);(2,4)(17,19,22,24,21,26,29,18)(20,25,28,30,27,31,32,23);(5,6,8,7)(10,13)(14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(5,7,8,6)(9,11,12,15)(10,14,13,16)(20,27)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300764);