#129362

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

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	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,f^b=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, 36558);