#405038

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

852 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 228 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=ef,c^2=e,d^2=gi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312234);