#498832

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fi,h^b=hi,e^c=ei,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, 406028);