#137708

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 455 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) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dgi,f^b=fgi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44904);