#359666

order	512
nilpotency class	3
isoclinism	isoclinic class of #359665
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	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3709 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)(4,20,23,35)(5,24)(6,27)(7,12)(8,13)(9,17)(10,22,36,29)(11,25)(15,30,44,33)(16,31)(18,32)(19,39)(21,51,54,34)(26,37)(28,38)(40,43,49,48)(41,45)(42,61,64,60)(46,57)(47,56)(50,53,59,52)(55,63,58,62);(1,3,8,19)(2,9,13,32)(4,15,23,44)(5,16,26,46)(6,18,7,17)(10,33,36,30)(11,31,38,57)(12,39,27,14)(20,49,35,40)(21,42,54,64)(22,48,29,43)(24,41,37,56)(25,47,28,45)(34,58,51,55)(50,61,59,60)(52,63,53,62);(1,4)(2,10)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30)(11,34)(12,35)(13,36)(14,40)(16,42)(17,43)(18,48)(19,44)(20,27)(24,50)(25,52)(26,54)(28,53)(31,55)(32,33)(37,59)(38,51)(39,49)(41,60)(45,62)(46,64)(47,63)(56,61)(57,58);(1,5)(2,11)(3,16)(4,21)(6,28)(7,25)(8,26)(9,31)(10,34)(12,37)(13,38)(14,41)(15,42)(17,47)(18,45)(19,46)(20,50)(22,53)(23,54)(24,27)(29,52)(30,55)(32,57)(33,58)(35,59)(36,51)(39,56)(40,60)(43,63)(44,64)(48,62)(49,61)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303489, #72310794, #72310890
maximal normal subgroup of	#72303489, #72310794, #72310890

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266862);