#494050

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#418
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20733
prime radical	#3584	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

911 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)(3,14,18,9)(4,20)(5,11)(6,26)(7,30)(8,12)(10,21)(13,29)(15,43,45,61)(16,34,48,33)(17,49,27,58)(19,35,51,42)(22,53)(23,55)(24,56)(25,39)(28,40)(31,36)(32,46,41,44)(37,57)(38,54)(47,63,52,60)(50,62,59,64);(1,3)(2,9)(4,15)(5,16)(6,27)(7,19)(8,18)(10,32)(11,33)(12,14)(13,35)(17,24)(20,43)(21,44)(22,45)(23,47)(25,48)(26,49)(28,59)(29,51)(30,42)(31,52)(34,39)(36,60)(37,62)(38,41)(40,64)(46,54)(50,57)(53,61)(55,63)(56,58);(1,4)(2,10)(3,15)(5,21)(6,28)(7,31)(8,22)(9,32)(11,36)(12,38)(13,40)(14,41)(16,44)(17,50)(18,45)(19,52)(20,26)(23,29)(24,57)(25,54)(27,59)(30,37)(33,60)(34,63)(35,64)(39,55)(42,62)(43,49)(46,48)(47,51)(53,56)(58,61);(1,5,7,24,8,25,29,6)(2,11,30,56,12,39,13,26)(3,16,19,17,18,48,51,27)(4,21,31,57,22,54,23,28)(9,33,42,58,14,34,35,49)(10,36,37,53,38,55,40,20)(15,44,52,50,45,46,47,59)(32,60,62,61,41,63,64,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320704, #72320710, #72320748
maximal normal subgroup of	#72320704, #72320710, #72320748

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401246);