#7514359

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2550 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,26)(13,31)(14,20)(15,35)(16,21)(18,29)(19,33)(22,43)(23,53)(24,40)(25,42)(27,38)(28,44)(30,45)(32,48)(34,49)(36,41)(37,46)(39,54)(47,56)(50,63)(51,64)(52,66)(55,65)(57,60)(58,72)(59,67)(61,69)(62,68)(70,71);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,21,42)(12,27,30,51)(13,23,34,55)(14,24,15,36)(18,43,46,66)(19,47,48,39)(20,40,35,41)(22,37,52,29)(26,38,45,64)(28,59,61,58)(31,53,49,65)(32,54,33,56)(44,67,69,72)(50,62,71,60)(57,63,68,70);(9,12)(10,18)(11,22)(13,28)(14,29)(15,37)(16,30)(17,38)(19,44)(20,45)(21,46)(23,50)(24,51)(25,52)(26,35)(27,36)(31,57)(32,60)(33,62)(34,61)(39,63)(40,43)(41,66)(42,64)(47,70)(48,69)(49,68)(53,67)(54,58)(55,71)(56,59)(65,72);(9,13,16,34)(10,19,21,48)(11,23,25,55)(12,28,30,61)(14,33,15,32)(17,39,42,47)(18,44,46,69)(20,49,35,31)(22,50,52,71)(24,56,36,54)(26,57,45,68)(27,58,51,59)(29,62,37,60)(38,63,64,70)(40,53,41,65)(43,67,66,72);(2,6)(5,7)(9,14,16,15)(10,20,21,35)(11,24,25,36)(12,29,30,37)(13,32,34,33)(17,40,42,41)(18,45,46,26)(19,31,48,49)(22,51,52,27)(23,54,55,56)(28,60,61,62)(38,43,64,66)(39,65,47,53)(44,57,69,68)(50,58,71,59)(63,72,70,67)

Polycyclic presentation:

2^9|b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi

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, 7421555);