#135038

order	512
nilpotency class	4
isoclinism	isoclinic class of #135037
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

272 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=i,d^2=gi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42234);