#498861

order	512
nilpotency class	4
isoclinism	isoclinic class of #498860
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1526 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)(7,18)(8,24)(9,30)(10,16)(11,36)(12,17)(13,21)(14,26)(15,34)(19,37)(20,49)(22,43)(23,56)(25,40)(27,44)(28,63)(29,59)(31,46)(32,47)(33,45)(35,58)(38,53)(39,55)(41,61)(42,62)(48,51)(50,64)(52,66)(54,65)(57,60)(67,68);(2,4)(5,7)(6,13)(8,25)(9,31)(10,23)(11,22)(12,20)(14,40)(15,43)(16,46)(17,38)(18,30)(19,27)(21,34)(24,57)(26,51)(28,50)(29,52)(32,60)(33,53)(35,55)(36,39)(37,41)(42,66)(44,48)(45,56)(47,64)(49,58)(54,61)(59,65)(62,68)(63,67);(5,8,12,29)(6,14,17,42)(7,19,20,50)(9,32,35,61)(10,28,11,27)(13,37,38,64)(15,44,45,63)(16,47,36,41)(18,48,49,67)(21,54,53,60)(22,52,23,25)(24,58,59,30)(26,33,62,34)(31,51,55,68)(39,66,46,40)(43,65,56,57);(7,20)(13,38)(18,49)(19,50)(21,53)(22,23)(25,52)(31,55)(37,64)(39,46)(40,66)(43,56)(48,67)(51,68)(54,60)(57,65)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281717, #72281854, #72282251
maximal normal subgroup of	#72281717, #72281854, #72282251

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406057);