#498808

order	512
nilpotency class	4
isoclinism	isoclinic class of #498807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1478 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)(7,19)(8,25)(9,30)(10,18)(11,17)(12,15)(13,21)(14,28)(16,33)(20,50)(22,44)(23,49)(24,48)(26,47)(27,58)(29,46)(31,38)(32,43)(34,45)(35,41)(36,57)(37,54)(39,53)(40,51)(42,61)(52,67)(55,65)(56,60)(59,64)(62,66)(63,68);(5,7,12,24)(6,13,15,37)(8,26,29,55)(9,31,34,53)(10,23,11,22)(14,40,41,64)(16,44,43,49)(17,39,18,38)(19,45,48,30)(20,35,52,28)(21,32,54,33)(25,36,46,63)(27,59,61,51)(42,65,58,47)(50,66,67,56)(57,62,68,60);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,28)(12,29)(13,36)(15,41)(16,42)(17,46)(18,25)(19,47)(21,51)(22,55)(23,26)(24,52)(30,56)(31,57)(32,62)(33,60)(34,61)(37,63)(38,64)(39,40)(43,58)(44,50)(45,66)(48,65)(49,67)(53,68)(54,59);(6,15)(13,37)(14,41)(16,43)(17,18)(19,48)(25,46)(30,45)(36,63)(38,39)(40,64)(42,58)(44,49)(47,65)(50,67)(56,66)

Polycyclic presentation:

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

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, 406004);