#394599

order	512
nilpotency class	3
isoclinism	isoclinic class of #394598
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2078 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)(3,13)(4,18)(5,10)(6,25)(7,12)(8,16)(9,22)(11,27)(14,36)(15,37)(17,35)(19,45)(20,31)(21,32)(23,51)(24,34)(26,30)(28,39)(29,40)(33,53)(38,59)(41,57)(42,47)(43,49)(44,50)(46,58)(48,62)(52,55)(54,63)(56,64)(60,61);(1,3)(2,8)(4,19)(5,15)(6,26)(7,17)(9,28)(10,29)(11,35)(12,30)(13,25)(14,22)(16,27)(18,42)(20,46)(21,39)(23,52)(24,41)(31,54)(32,45)(33,57)(34,55)(36,44)(37,51)(38,49)(40,53)(43,61)(47,50)(48,60)(56,59)(58,62)(63,64);(1,4,6,21,7,22,27,50)(2,9,11,32,12,18,25,44)(3,14,16,39,17,19,26,47)(5,20,23,48,24,49,53,64)(8,28,13,36,30,42,35,45)(10,31,33,56,34,43,51,62)(15,38,40,60,41,46,52,63)(29,54,37,58,55,61,57,59);(1,5,7,24)(2,10,12,34)(3,15,17,41)(4,20,22,49)(6,23,27,53)(8,29,30,55)(9,31,18,43)(11,33,25,51)(13,37,35,57)(14,38,19,46)(16,40,26,52)(21,48,50,64)(28,54,42,61)(32,56,44,62)(36,58,45,59)(39,60,47,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317848, #72317961, #72318571
maximal normal subgroup of	#72317848, #72317961, #72318571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301795);