#513764

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420960);