#7622939

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2030 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)(5,6,12,18)(7,13,20,35)(8,24,27,56)(9,17,11,15)(10,29,32,42)(14,39,41,26)(16,28,44,31)(19,46,47,66)(21,38,23,36)(22,52,50,63)(25,57,33,43)(30,45,58,40)(34,59,60,49)(37,51,62,54)(48,67,55,64)(53,65,68,61);(1,2,3,4)(5,7)(6,13)(8,19)(9,21)(10,22)(11,23)(12,20)(14,34)(15,36)(16,37)(17,38)(18,35)(24,46)(25,48)(26,49)(27,47)(28,51)(29,52)(30,53)(31,54)(32,50)(33,55)(39,59)(40,61)(41,60)(42,63)(43,64)(44,62)(45,65)(56,66)(57,67)(58,68);(5,8)(6,14)(7,19)(9,25)(10,30)(11,33)(12,27)(13,34)(15,40)(16,43)(17,45)(18,41)(20,47)(21,48)(22,53)(23,55)(24,29)(26,31)(28,39)(32,58)(35,60)(36,61)(37,64)(38,65)(42,56)(44,57)(46,52)(49,54)(50,68)(51,59)(62,67)(63,66);(7,20)(13,35)(19,47)(21,23)(22,50)(34,60)(36,38)(37,62)(46,66)(48,55)(49,59)(51,54)(52,63)(53,68)(61,65)(64,67);(5,9,12,11)(6,15,18,17)(7,21,20,23)(8,25,27,33)(10,28,32,31)(13,36,35,38)(14,40,41,45)(16,42,44,29)(19,48,47,55)(22,51,50,54)(24,43,56,57)(26,30,39,58)(34,61,60,65)(37,63,62,52)(46,64,66,67)(49,53,59,68)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343342, #72343350, #72343374, …
maximal normal subgroup of	#72343342, #72343350, #72343374, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530135);