#494457

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,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, 401653);