#7622498

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×#578	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3912 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529694);