#413937

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15159
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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)(5,25)(6,28)(7,12)(8,13)(9,17)(11,27)(15,33)(16,42)(18,39)(19,32)(20,49)(21,34)(22,40)(23,35)(24,36)(26,37)(29,38)(30,44)(31,48)(41,45)(43,61)(46,54)(47,57)(50,56)(51,60)(52,55)(53,59)(58,62)(63,64);(1,3)(2,9)(4,20)(5,16)(6,19)(7,17)(8,18)(10,33)(11,31)(12,39)(13,32)(14,28)(15,23)(21,50)(22,45)(24,44)(25,54)(26,48)(27,47)(29,46)(30,35)(34,58)(36,41)(37,42)(38,57)(40,49)(43,51)(52,63)(53,62)(55,61)(56,60)(59,64);(1,4)(2,10)(3,15)(5,21)(6,23)(7,22)(8,24)(9,30)(11,34)(12,40)(13,36)(14,41)(16,43)(17,20)(18,45)(19,44)(25,55)(26,53)(27,52)(28,35)(29,51)(31,56)(32,49)(33,39)(37,60)(38,59)(42,61)(46,63)(47,62)(48,50)(54,64)(57,58);(1,5,6,29,8,27,7,26)(2,11,28,37,13,25,12,38)(3,16,19,46,18,47,17,48)(4,21,23,51,24,52,22,53)(9,31,14,42,32,54,39,57)(10,34,35,60,36,55,40,59)(15,43,44,63,45,62,20,50)(30,56,41,61,49,64,33,58)

Polycyclic presentation:

2^9|d^2=gh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 321133);