#137707

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2)(3,4)(5,6,10,11,9,12,8,7)(13,14,17,20,18,21,25,27,16,15,19,22,24,23,26,28);(5,7,10,12,9,11,8,6)(13,15,17,22,18,23,25,28,16,14,19,20,24,21,26,27);(2,4)(5,8,9,10)(6,11,12,7)(14,15)(20,22)(21,23)(27,28)

Polycyclic presentation:

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

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, 44903);