#129776

order	512
nilpotency class	4
isoclinism	isoclinic class of #124648
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36972);