#147875

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1120 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:

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

Polycyclic presentation:

2^9|a^2=g,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=ehi,h^b=hi,d^c=dhi,e^c=eh,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, 55071);