#7623107

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1490 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)(9,10,16,22)(11,17,24,39)(12,28,31,60)(13,21,15,19)(14,33,36,46)(18,43,45,30)(20,32,48,35)(23,50,51,70)(25,42,27,40)(26,56,54,67)(29,61,37,47)(34,49,62,44)(38,63,64,53)(41,55,66,58)(52,71,59,68)(57,69,72,65);(2,5)(6,8)(9,11,16,24)(10,17,22,39)(12,23,31,51)(13,25,15,27)(14,26,36,54)(18,38,45,64)(19,40,21,42)(20,41,48,66)(28,50,60,70)(29,52,37,59)(30,53,43,63)(32,55,35,58)(33,56,46,67)(34,57,62,72)(44,65,49,69)(47,68,61,71);(9,12)(10,18)(11,23)(13,29)(14,34)(15,37)(16,31)(17,38)(19,44)(20,47)(21,49)(22,45)(24,51)(25,52)(26,57)(27,59)(28,33)(30,35)(32,43)(36,62)(39,64)(40,65)(41,68)(42,69)(46,60)(48,61)(50,56)(53,58)(54,72)(55,63)(66,71)(67,70);(1,3,4,7)(2,6,5,8)(11,24)(17,39)(23,51)(25,27)(26,54)(38,64)(40,42)(41,66)(50,70)(52,59)(53,63)(55,58)(56,67)(57,72)(65,69)(68,71);(9,13,16,15)(10,19,22,21)(11,25,24,27)(12,29,31,37)(14,32,36,35)(17,40,39,42)(18,44,45,49)(20,46,48,33)(23,52,51,59)(26,55,54,58)(28,47,60,61)(30,34,43,62)(38,65,64,69)(41,67,66,56)(50,68,70,71)(53,57,63,72)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530303);