#414981

order	512
nilpotency class	3
isoclinism	isoclinic class of #403928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2615
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15181
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1219 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:

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

Polycyclic presentation:

2^9|a^2=efh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=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, 322177);