#458576

order	512
nilpotency class	3
isoclinism	isoclinic class of #458575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16320
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 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)(3,8,15,29)(4,9)(5,10)(6,23)(7,11)(12,26)(13,40,42,27)(14,28,44,55)(16,38,24,30)(17,31,48,59)(18,35)(19,51)(20,33)(21,39)(22,36)(25,37)(32,50)(34,53)(41,54,43,63)(45,57,46,56)(47,62,64,58)(49,60)(52,61);(1,3)(2,8,11,29)(4,17)(5,14)(6,24)(7,15)(9,31,33,59)(10,28,36,55)(12,38,23,30)(13,19)(16,26)(18,47)(20,48)(21,46)(22,44)(25,45)(27,51,40,34)(32,58,35,62)(37,57,39,56)(41,52)(42,53)(43,49)(50,64)(54,60,63,61);(1,4)(2,9)(3,13,15,42)(5,18)(6,19)(7,20)(8,27,29,40)(10,35)(11,33)(12,34)(14,43,44,41)(16,17,24,48)(21,49)(22,50)(23,51)(25,52)(26,53)(28,54,55,63)(30,31,38,59)(32,36)(37,61)(39,60)(45,47,46,64)(56,62,57,58);(1,5,7,22)(2,10,11,36)(3,14,15,44)(4,18,20,50)(6,25,26,21)(8,28,29,55)(9,32,33,35)(12,39,23,37)(13,41,42,43)(16,46,24,45)(17,47,48,64)(19,52,53,49)(27,54,40,63)(30,57,38,56)(31,58,59,62)(34,61,51,60)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=i,g^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,g^b=gi,d^c=de,h^c=hi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285845, #72286106, #72286186
maximal normal subgroup of	#72285845, #72286106, #72286186

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365772);