#393386

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

1162 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)(12,16)(13,15)(17,18,22,25,21,24,26,19)(20,28,31,27,30,23,32,29);(5,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,26,24,21,25,22,18)(20,29,32,23,30,27,31,28);(9,12,13,11)(10,14,15,16)(17,20,21,30)(18,23,24,28)(19,27,25,29)(22,31,26,32);(2,4)(5,7,8,6)(18,24)(19,25)(20,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci,d^b=di,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, 300582);