#6682665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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

2108 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)(9,10,16,22)(11,17,27,42)(12,28,32,44)(13,19,15,21)(14,20,36,50)(18,37,46,30)(23,43,53,65)(24,39,26,41)(25,40,55,64)(29,56,52,38)(31,57,61,67)(33,47,35,49)(34,48,54,63)(45,60,68,59)(51,66,70,72)(58,71,69,62);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,29)(13,26)(14,34)(15,24)(16,27)(18,43)(19,41)(20,48)(21,39)(22,42)(23,30)(25,33)(28,56)(31,51)(32,52)(35,55)(36,54)(37,53)(38,44)(40,47)(45,62)(46,65)(49,64)(50,63)(57,66)(58,60)(59,69)(61,70)(67,72)(68,71);(9,12,15,37,16,32,13,30)(10,18,19,44,22,46,21,28)(11,23,24,29,27,53,26,52)(14,31,35,60,36,61,33,59)(17,38,41,65,42,56,39,43)(20,45,47,67,50,68,49,57)(25,51,34,58,55,70,54,69)(40,62,63,72,64,71,48,66);(2,6)(5,7)(9,13,16,15)(10,19,22,21)(11,24,27,26)(12,30,32,37)(14,35,36,33)(17,39,42,41)(18,44,46,28)(20,49,50,47)(23,29,53,52)(25,54,55,34)(31,60,61,59)(38,43,56,65)(40,63,64,48)(45,57,68,67)(51,69,70,58)(62,72,71,66);(3,8)(5,7)(9,14)(10,20)(11,25)(12,31)(13,33)(15,35)(16,36)(17,40)(18,45)(19,47)(21,49)(22,50)(23,51)(24,34)(26,54)(27,55)(28,57)(29,58)(30,59)(32,61)(37,60)(38,62)(39,48)(41,63)(42,64)(43,66)(44,67)(46,68)(52,69)(53,70)(56,71)(65,72)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589861);