#6370756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370755
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#379
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×2₋¹⁺⁶	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

16831 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)(4,18,21,9)(5,10)(6,24)(7,12)(8,16)(11,26)(14,37,40,35)(15,36)(17,34)(19,44,46,30)(20,31,49,45)(22,32)(23,51)(25,29)(27,39,43,48)(28,42)(33,50)(38,58,59,57)(41,56)(47,55,63,62)(52,54)(53,60,61,64);(1,3,7,17)(2,8,12,29)(4,14,21,40)(5,15,22,41)(6,25,26,16)(9,27,18,43)(10,28,32,54)(11,34,24,13)(19,38,46,59)(20,48,49,39)(23,52,50,42)(30,53,44,61)(31,37,45,35)(33,56,51,36)(47,64,63,60)(55,58,62,57);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,27)(10,30)(11,31)(12,18)(13,35)(15,38)(16,39)(17,40)(22,46)(23,47)(24,45)(25,48)(26,49)(28,53)(29,43)(32,44)(33,55)(34,37)(36,57)(41,59)(42,60)(50,63)(51,62)(52,64)(54,61)(56,58);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,28)(9,30)(11,33)(12,32)(13,36)(14,38)(16,42)(17,41)(18,44)(20,47)(21,46)(24,51)(25,52)(26,50)(27,53)(29,54)(31,55)(34,56)(35,57)(37,58)(39,60)(40,59)(43,61)(45,62)(48,64)(49,63);(5,22)(10,32)(15,41)(19,46)(23,50)(28,54)(30,44)(33,51)(36,56)(38,59)(42,52)(47,63)(53,61)(55,62)(57,58)(60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310794, #72310890, #72310944
maximal normal subgroup of	#72310794, #72310890, #72310944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277952);