#498830

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

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,f^b=fi,h^b=hi,e^c=ei,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, 406026);