#383163

order	512
nilpotency class	3
isoclinism	isoclinic class of #383160
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14357
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1887 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319923, #72319962, #72320118
maximal normal subgroup of	#72319923, #72319962, #72320118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290359);