#138338

order	512
nilpotency class	4
isoclinism	isoclinic class of #125508
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

630 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 102 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)(13,14,17,22,34,36,31,21,18,23,32,41,19,24,16,30)(15,25,28,20,35,38,27,42,26,43,44,37,29,40,33,39);(5,7,8,11)(6,9)(10,12)(13,15,19,35,18,26,34,29)(14,20,36,39,23,37,24,42)(16,28,32,27,31,44,17,33)(21,40,41,43,30,38,22,25);(2,4)(7,11)(9,12)(15,26)(20,37)(25,43)(27,33)(28,44)(29,35)(38,40)(39,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45534);