#141020

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

368 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,d^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48216);