#304942

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=efh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212138);