#393515

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1866 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)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(20,23)(24,28)(25,29)(27,30)(31,32);(6,8)(17,18,23,29,21,25,20,22)(19,24,30,32,26,31,27,28);(10,13)(14,16)(17,19)(18,24)(20,27)(21,26)(22,28)(23,30)(25,31)(29,32);(2,4)(9,11,12,15)(10,14,13,16)(19,26)(24,31)(27,30)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300711);