#135156

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	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

716 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,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, 42352);