#135986

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

453 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

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

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, 43182);