#7622934

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343750, #72343825, #72343827
maximal normal subgroup of	#72343750, #72343825, #72343827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530130);