#417668

order	512
nilpotency class	3
isoclinism	isoclinic class of #417667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15257
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60398	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2329 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)(4,10)(5,11)(6,27)(7,12)(8,13)(9,17)(15,43)(16,44)(18,32)(19,42)(20,30)(21,51)(22,36)(23,37)(24,55)(25,41)(26,40)(28,39)(29,35)(31,48)(33,46)(34,53)(38,47)(45,59)(49,58)(50,60)(52,61)(54,62)(56,57)(63,64);(1,3,8,19)(2,9,13,32)(4,20,23,49)(5,24,26,48)(6,18,7,17)(10,33,37,43)(11,38,40,44)(12,42,27,14)(15,29,46,22)(16,28,47,25)(21,45,53,64)(30,35,58,36)(31,41,55,39)(34,60,51,57)(50,54,56,52)(59,62,63,61);(1,4)(2,10)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30)(11,34)(12,35)(13,37)(14,43)(16,45)(17,20)(18,49)(19,46)(24,56)(25,52)(26,53)(27,36)(28,54)(31,57)(32,58)(33,42)(38,59)(39,61)(40,51)(41,62)(44,63)(47,64)(48,50)(55,60);(1,5)(2,11)(3,16)(4,21)(6,28)(7,25)(8,26)(9,31)(10,34)(12,41)(13,40)(14,44)(15,45)(17,48)(18,24)(19,47)(20,50)(22,54)(23,53)(27,39)(29,52)(30,57)(32,55)(33,59)(35,62)(36,61)(37,51)(38,42)(43,63)(46,64)(49,56)(58,60)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324864);