#147899

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55095);