#498811

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1504 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)(7,19)(8,24)(9,15)(10,30)(11,17)(12,18)(13,22)(14,27)(16,32)(20,48)(21,46)(23,47)(25,40)(26,57)(28,44)(29,43)(31,36)(33,42)(34,41)(35,62)(37,61)(38,53)(39,60)(45,51)(49,67)(50,54)(52,65)(55,59)(56,66)(58,64)(63,68);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,31)(11,34)(12,23)(14,35)(15,36)(16,41)(17,37)(18,38)(19,30)(22,32)(24,54)(25,49)(26,58)(27,51)(28,52)(29,53)(33,61)(39,65)(40,64)(42,47)(43,46)(44,63)(45,55)(48,57)(50,59)(56,68)(60,62)(66,67);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,35)(15,24)(16,39)(17,44)(18,40)(19,45)(21,49)(22,50)(23,52)(29,56)(30,55)(31,58)(32,59)(33,60)(34,51)(36,54)(37,63)(38,64)(41,65)(42,66)(43,57)(46,48)(47,67)(53,68)(61,62);(5,9,12,11)(6,15,18,17)(7,21,23,34)(8,25,28,27)(10,29,33,32)(13,36,38,37)(14,24,40,44)(16,30,43,42)(19,46,47,41)(20,49,52,51)(22,31,53,61)(26,56,60,59)(35,54,64,63)(39,55,57,66)(45,48,67,65)(50,58,68,62)

Polycyclic presentation:

2^9|d^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,e^b=eh,h^b=hi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406007);