#405281

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14911
prime radical	#60398	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2176 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)(6,26)(7,12)(8,13)(9,17)(15,33)(16,40)(18,31)(19,46)(20,48)(21,36)(22,37)(23,51)(24,39)(25,38)(27,32)(28,35)(29,42)(30,44)(34,50)(41,61)(43,57)(45,56)(47,54)(49,59)(52,55)(53,60)(58,63)(62,64);(1,3)(2,9)(4,19,22,42)(5,16)(6,27)(7,17)(8,18)(10,33,37,57)(11,30)(12,31)(13,32)(14,26)(15,21,43,28)(20,47,50,62)(23,52)(24,45)(25,44)(29,36,46,35)(34,58,48,61)(38,56)(39,55)(40,51)(41,49,63,53)(54,60,64,59);(1,4)(2,10)(3,15)(5,20)(6,21)(7,28)(8,22)(9,29)(11,34)(12,35)(13,37)(14,33)(16,41)(17,42)(18,43)(19,27)(23,49)(24,50)(25,53)(26,36)(30,54)(31,57)(32,46)(38,59)(39,48)(40,58)(44,62)(45,63)(47,52)(51,60)(55,64)(56,61);(1,5)(2,11)(3,16)(4,20)(6,23)(7,25)(8,24)(9,30)(10,34)(12,38)(13,39)(14,40)(15,41)(17,44)(18,45)(19,47)(21,49)(22,50)(26,51)(27,52)(28,53)(29,54)(31,56)(32,55)(33,58)(35,59)(36,60)(37,48)(42,62)(43,63)(46,64)(57,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312477);