#364546

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13967
prime radical	#60398	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2343 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317342, #72317353, #72317361
maximal normal subgroup of	#72317342, #72317353, #72317361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271742);