#147892

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
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:

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

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,h^a=hi,c^b=cf,d^b=dh,e^b=eh,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, 55088);