#499508

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23697
prime radical	#13312	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320718, #72320727, #72320748
maximal normal subgroup of	#72320718, #72320727, #72320748

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406704);