#499085

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

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

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, 406281);