#135217

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=ei,f^c=fgi

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, 42413);