#418482

order	512
nilpotency class	3
isoclinism	isoclinic class of #418482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15263
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319923, #72319954, #72320088
maximal normal subgroup of	#72319923, #72319954, #72320088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325678);