#7623279

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1356 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)(11,22)(12,26)(13,17)(14,19)(15,36)(16,37)(18,30)(20,27)(21,35)(23,47)(24,51)(25,40)(28,52)(29,53)(31,63)(32,64)(33,46)(34,65)(38,59)(39,57)(41,61)(42,67)(43,58)(44,49)(45,48)(50,71)(54,56)(55,62)(60,69)(66,70)(68,72);(2,5)(6,8)(9,11,14,13)(10,17,19,22)(12,23,29,28)(15,24,34,32)(16,25,35,33)(18,38,41,44)(20,39,43,45)(21,40,37,46)(26,52,53,47)(27,48,58,57)(30,49,61,59)(31,50,62,60)(36,64,65,51)(42,66,54,68)(55,71,63,69)(56,70,67,72);(9,12,27,56,16,31,34,61,14,29,58,67,35,62,15,30)(10,18,36,55,21,42,43,53,19,41,65,63,37,54,20,26)(11,23,48,70,25,50,32,59,13,28,57,72,33,60,24,49)(17,38,64,71,40,66,45,47,22,44,51,69,46,68,39,52);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,17,19,22)(12,28,29,23)(15,32,34,24)(16,33,35,25)(18,38,41,44)(20,39,43,45)(21,40,37,46)(26,52,53,47)(27,57,58,48)(30,59,61,49)(31,60,62,50)(36,64,65,51)(42,66,54,68)(55,71,63,69)(56,72,67,70);(11,13)(12,29)(17,22)(18,41)(24,32)(25,33)(26,53)(30,61)(31,62)(39,45)(40,46)(42,54)(48,57)(51,64)(55,63)(56,67)

Polycyclic presentation:

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

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, 7530475);