#401011

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1638 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 308207);