#137976

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 662 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 45172);