#416559

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323755);