#124299

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×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

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

normal subgroups only

This group has 326 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(2,5,6,7)(10,14)(11,12)(15,16)(17,18,21,27,23,28,36,26,22,25,37,42,35,43,20,34)(19,29,31,39,33,44,47,46,32,45,48,24,38,40,30,41);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19,22,32)(18,24,25,39)(20,31,36,48)(21,30,37,47)(23,38,35,33)(26,44,34,40)(27,45,42,29)(28,46,43,41);(9,11,13,12)(10,14,15,16)(18,25)(24,39)(26,34)(27,42)(28,43)(29,45)(40,44)(41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31495);