#131830

order	512
nilpotency class	4
isoclinism	isoclinic class of #131824
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	#61477	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)(9,10,15,22,17,23,16,38)(11,24,28,54,29,55,39,42)(12,30,33,51,34,66,40,46)(13,21,35,50,36,52,14,20)(18,27,43,62,45,64,44,26)(19,31,47,68,49,37,48,32)(25,56,59,70,61,71,60,41)(53,63,69,67,72,57,65,58);(1,2)(3,7)(4,6)(5,8)(9,11,17,29)(10,18,23,45)(12,25,34,61)(13,27,36,64)(14,26,35,62)(15,28,16,39)(19,41,49,70)(20,42,50,54)(21,24,52,55)(22,43,38,44)(30,65,66,69)(31,67,37,58)(32,63,68,57)(33,59,40,60)(46,72,51,53)(47,56,48,71);(9,12)(10,19)(11,25)(13,32)(14,37)(15,33)(16,40)(17,34)(18,41)(20,30)(21,51)(22,47)(23,49)(24,53)(26,58)(27,63)(28,59)(29,61)(31,35)(36,68)(38,48)(39,60)(42,65)(43,56)(44,71)(45,70)(46,52)(50,66)(54,69)(55,72)(57,64)(62,67)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=g,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=egi,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, 39026);