#359537

order	512
nilpotency class	3
isoclinism	isoclinic class of #359534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13867
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62848	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3699 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303163, #72303351, #72303359
maximal normal subgroup of	#72303163, #72303351, #72303359

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266733);