#124475

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=fgh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi

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, 31671);