#147885

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
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

1110 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)(7,17)(8,22)(9,27)(10,33)(11,15)(12,19)(13,24)(14,31)(16,30)(18,36)(20,58)(21,50)(23,41)(25,63)(26,40)(28,46)(29,47)(32,48)(34,43)(35,44)(37,57)(38,64)(39,54)(42,62)(45,55)(49,52)(51,60)(53,68)(56,66)(59,67)(61,65);(5,7,11,21)(6,12,15,37)(8,18,26,54)(9,28,32,64)(10,34,30,20)(13,39,41,36)(14,43,45,58)(16,46,33,38)(17,48,50,27)(19,55,57,31)(22,60,40,49)(23,61,24,56)(25,59,35,53)(29,52,62,51)(42,67,47,68)(44,65,63,66);(5,8)(6,13)(7,18)(9,29)(10,35)(11,26)(12,36)(14,44)(15,41)(16,47)(17,49)(19,56)(20,59)(21,54)(22,27)(23,55)(24,31)(25,30)(28,52)(32,62)(33,42)(34,53)(37,39)(38,68)(40,48)(43,66)(45,63)(46,67)(50,60)(51,64)(57,61)(58,65)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,h^b=hi,d^c=dhi,e^c=eh,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, 55081);