#128532

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,h^a=hi,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35728);