#413903

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6)(7,12)(8,13)(9,26)(10,15)(11,17)(14,28)(16,31)(18,32)(19,46)(20,34)(21,36)(22,37)(23,51)(24,39)(25,41)(27,43)(29,35)(30,40)(33,48)(38,53)(42,56)(44,65)(45,61)(47,54)(49,60)(50,59)(52,55)(57,62)(58,64)(63,66)(67,68);(2,4)(5,7,11,21)(6,12,17,36)(8,22,25,49)(9,19,27,47)(10,29,31,20)(13,37,41,60)(14,33,42,62)(15,35,16,34)(18,30,45,24)(23,50,52,67)(26,54,43,46)(28,57,56,48)(32,40,61,39)(38,63,64,65)(44,58,66,53)(51,68,55,59);(5,8,11,25)(6,13,17,41)(7,18,21,45)(9,23,27,52)(10,30,31,24)(12,32,36,61)(14,38,42,64)(15,40,16,39)(19,44,47,66)(20,49,29,22)(26,55,43,51)(28,58,56,53)(33,59,62,68)(34,60,35,37)(46,63,54,65)(48,67,57,50);(5,9,11,27)(6,14)(7,19,21,47)(8,23,25,52)(10,28,31,56)(12,33)(13,38)(15,43)(16,26)(17,42)(18,44,45,66)(20,48,29,57)(22,50,49,67)(24,53,30,58)(32,59)(34,54)(35,46)(36,62)(37,63)(39,55)(40,51)(41,64)(60,65)(61,68)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,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, 321099);