#129766

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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×#3050	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 956 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) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36962);