#140905

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48101);