#495892

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403088);