#131823

order	512
nilpotency class	4
isoclinism	isoclinic class of #131820
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#61491	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 562 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5,6,7)(9,10,15,22,17,24,16,23)(11,25,29,53,31,55,30,44)(12,32,35,66,37,67,36,49)(13,38,40,20,41,21,14,42)(18,39,45,27,47,28,46,63)(19,48,50,34,52,33,51,68)(26,56,59,43,61,70,60,71)(54,57,72,58,69,64,65,62);(1,2)(3,7)(4,6)(5,8)(9,11)(10,18)(12,26)(13,39)(14,27)(15,30)(16,29)(17,31)(19,43)(20,53)(21,25)(22,46)(23,45)(24,47)(28,41)(32,65)(33,62)(34,57)(35,60)(36,59)(37,61)(38,55)(40,63)(42,44)(48,58)(49,54)(50,56)(51,70)(52,71)(64,68)(66,69)(67,72);(9,12,17,37)(10,19,24,52)(11,26,31,61)(13,34,41,68)(14,33,40,48)(15,36,16,35)(18,43,47,71)(20,32,42,67)(21,49,38,66)(22,51,23,50)(25,54,55,69)(27,62,63,58)(28,64,39,57)(29,60,30,59)(44,72,53,65)(45,56,46,70)

Polycyclic presentation:

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

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, 39019);