#410666

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1288 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)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,22,23,21,26,20,25)(19,27,30,32,29,24,28,31);(2,4)(5,6,8,7)(10,14)(15,16)(18,23)(19,28)(20,22)(25,26)(29,30)(31,32);(5,6,8,7)(9,11,13,12)(10,15,14,16)(17,19,20,28,21,29,22,30)(18,24,25,32,26,27,23,31);(5,7,8,6)(9,12,13,11)(10,15,14,16)(19,29)(24,27)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,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, 317862);