#142303

order	512
nilpotency class	4
isoclinism	isoclinic class of #142232
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 88 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)(8,10)(11,12)(13,14,20,26)(15,27,32,43)(16,23,34,40)(17,22,33,41)(18,25,19,24)(21,35,39,29)(28,37,44,38)(30,36,31,42);(9,12)(13,15,20,32)(14,21,26,39)(16,29,34,35)(17,28,33,44)(18,31,19,30)(22,27,41,43)(23,36,40,42)(24,38,25,37);(2,4)(5,7)(6,9)(8,11)(10,12)(13,16,19,33,20,34,18,17)(14,22,25,40,26,41,24,23)(15,28,31,35,32,44,30,29)(21,36,38,43,39,42,37,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^b=gi,d^c=dh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49499);