#7622454

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622451
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	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂×D₄	C₂⁵
prime radical	#62859	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4093 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)(3,13)(4,9)(5,10)(6,24,25,11)(7,12)(8,16,28,42)(14,35)(15,36)(17,37)(18,29)(19,30)(20,47,44,31)(21,48)(22,50,51,33)(23,34)(26,40,52,59)(27,41,32,49)(38,57)(39,58)(43,54)(45,62)(46,64,61,56)(53,60,55,63);(1,3,7,17)(2,8,12,28)(4,14,18,38)(5,21,23,15)(6,16,25,42)(9,26,29,52)(10,32,34,27)(11,13,24,37)(19,45,43,39)(20,40,44,59)(22,49,51,41)(30,55,54,53)(31,35,47,57)(33,48,50,36)(46,63,61,60)(56,62,64,58);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,26)(10,30)(11,31)(12,29)(13,35)(15,39)(16,40)(17,38)(21,45)(22,46)(23,43)(24,47)(25,44)(27,53)(28,52)(32,55)(33,56)(34,54)(36,58)(37,57)(41,60)(42,59)(48,62)(49,63)(50,64)(51,61);(4,18)(9,29)(14,38)(19,43)(20,44)(26,52)(30,54)(31,47)(35,57)(39,45)(40,59)(46,61)(53,55)(56,64)(58,62)(60,63);(1,5,7,23)(2,10,12,34)(3,15,17,21)(4,19,18,43)(6,22,25,51)(8,27,28,32)(9,30,29,54)(11,33,24,50)(13,36,37,48)(14,39,38,45)(16,41,42,49)(20,46,44,61)(26,53,52,55)(31,56,47,64)(35,58,57,62)(40,60,59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302919, #72302953, #72303046, …
maximal normal subgroup of	#72302919, #72302953, #72303046, …
maximal quotient of	#72312266, #72312798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529650);