#7623167

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
prime radical	#60643	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530363);