#7622773

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3529	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1287 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)(9,10,13,19,14,20,30,43,15,21,31,44,32,45,55,66)(11,16,24,34,25,35,50,59,22,17,36,38,47,39,60,63)(12,26,27,51,28,52,53,69,29,18,40,41,54,42,64,65)(23,37,48,61,49,62,68,72,46,33,56,57,67,58,70,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,23)(13,24)(14,25)(15,22)(17,21)(18,33)(19,34)(20,35)(26,37)(27,48)(28,49)(29,46)(30,50)(31,36)(32,47)(38,44)(39,45)(40,56)(41,57)(42,58)(43,59)(51,61)(52,62)(53,68)(54,67)(55,60)(63,66)(64,70)(65,71)(69,72);(3,8)(5,7)(9,11)(10,17)(12,23)(13,24)(14,25)(15,22)(16,21)(18,37)(19,38)(20,39)(26,33)(27,48)(28,49)(29,46)(30,50)(31,36)(32,47)(34,44)(35,45)(40,56)(41,61)(42,62)(43,63)(51,57)(52,58)(53,68)(54,67)(55,60)(59,66)(64,70)(65,72)(69,71);(2,6)(5,7)(11,22)(16,17)(23,46)(24,36)(25,47)(33,37)(34,38)(35,39)(48,56)(49,67)(50,60)(57,61)(58,62)(59,63)(68,70)(71,72);(9,12,13,27,14,28,30,53,15,29,31,40,32,54,55,64)(10,18,19,41,20,42,43,65,21,26,44,51,45,52,66,69)(11,23,24,48,25,49,50,68,22,46,36,56,47,67,60,70)(16,33,34,57,35,58,59,71,17,37,38,61,39,62,63,72)

Polycyclic presentation:

2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529969);