#499019

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	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1066 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)(8,24)(9,15)(10,31)(11,17)(12,18)(14,27)(16,34)(19,49)(20,37)(21,53)(22,48)(23,39)(25,57)(26,58)(28,45)(29,59)(30,47)(32,63)(33,41)(35,46)(36,50)(38,55)(40,60)(42,61)(43,62)(44,52)(51,67)(54,64)(56,66)(65,68);(1,2)(3,4)(5,7)(6,13)(8,25)(9,20)(10,32)(11,22)(12,23)(14,40)(15,37)(16,44)(17,48)(18,39)(19,28)(21,35)(24,57)(26,51)(27,60)(29,56)(30,55)(31,63)(33,50)(34,52)(36,41)(38,47)(42,65)(43,64)(45,49)(46,53)(54,62)(58,67)(59,66)(61,68);(1,3,4,2)(5,8)(6,14)(7,19)(9,26)(10,33)(11,28)(12,29)(13,36)(15,41)(16,45)(17,42)(18,43)(20,25)(21,54)(22,56)(23,51)(24,31)(27,34)(30,62)(32,60)(35,61)(37,64)(38,66)(39,65)(40,48)(44,57)(46,59)(47,58)(49,53)(50,55)(52,68)(63,67);(5,9,12,11)(6,15,18,17)(7,20,23,22)(8,26,29,28)(10,30,35,34)(13,37,39,48)(14,41,43,42)(16,31,47,46)(19,25,51,56)(21,52,32,55)(24,58,59,45)(27,33,62,61)(36,64,65,40)(38,53,44,63)(49,57,67,66)(50,54,68,60)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,g^c=gi,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, 406215);