#7622486

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622467
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂³×#99
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×Q₈	C₂⁵
prime radical	#62859	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

4073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529682);