#320227

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,12,16,13,15,21,23)(11,17,18,22,19,24,20,14)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(2,5)(6,7)(25,27,29,35,28,34,36,33)(26,30,32,37,31,38,39,40);(9,11,13,19)(10,14,15,22)(12,20,21,18)(16,24,23,17)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,23)(17,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227423);