#443948

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,e^d=ei

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, 351144);