#135201

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

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

normal subgroups only

This group has 572 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) = 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)(17,18,22,28,23,29,32,40,24,30,19,25,21,27,33,41)(20,34,31,46,35,43,47,45,36,44,38,48,37,26,42,39);(2,5)(6,8)(17,19,23,33,24,22,21,32)(18,25,29,41,30,28,27,40)(20,31,35,47,36,38,37,42)(26,39,34,46,43,45,44,48);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,37)(22,38)(23,35)(24,36)(25,39)(27,44)(28,45)(29,34)(30,43)(32,42)(33,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=f,b^2=h,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, 42397);