#499158

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1008 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)(8,23)(9,15)(10,31)(11,35)(12,17)(13,21)(14,27)(16,33)(19,50)(20,48)(22,49)(24,47)(25,55)(26,44)(28,61)(29,43)(30,46)(32,38)(34,63)(36,45)(37,51)(39,64)(40,54)(41,52)(42,60)(53,65)(56,59)(57,68)(58,66)(62,67);(5,7)(6,13)(8,24)(9,20)(10,32)(11,36)(12,22)(14,41)(15,38)(16,45)(17,40)(18,31)(19,27)(21,33)(23,37)(25,57)(26,53)(28,62)(29,52)(30,54)(34,64)(35,39)(42,67)(43,66)(44,47)(46,48)(49,63)(50,61)(51,60)(55,65)(56,68)(58,59);(2,4)(5,8,10,26,9,25,30,59,12,29,34,27,11,28,33,60)(6,14,16,43,35,56,63,55,17,44,46,23,15,42,31,61)(7,19,21,52,36,58,64,57,22,53,54,24,20,51,32,62)(13,37,18,47,38,65,48,68,40,66,49,41,39,50,45,67);(5,9,12,11)(6,15,17,35)(7,20,22,36)(8,25,29,28)(10,30,34,33)(13,38,40,39)(14,42,44,56)(16,31,46,63)(18,48,49,45)(19,51,53,58)(21,32,54,64)(23,55,43,61)(24,57,52,62)(26,59,27,60)(37,65,66,50)(41,67,47,68)

Polycyclic presentation:

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

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, 406354);