#147951

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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

954 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)(7,19,24,50)(8,26)(9,17)(10,34)(11,37)(12,18)(13,21,41,59)(14,29)(15,32)(16,36)(20,54,57,61)(22,40,25,47)(23,39,60,42)(27,62,64,67)(28,51)(30,48)(31,49)(33,52)(35,46)(38,63,68,55)(43,56,53,58)(44,66)(45,65);(5,7)(6,13)(8,27)(9,23)(10,22)(11,21)(12,24)(14,43)(15,47)(16,39)(17,50)(18,41)(19,52)(20,30)(25,36)(26,61)(28,56)(29,63)(31,64)(32,59)(33,60)(34,42)(35,55)(37,40)(38,45)(44,62)(46,53)(48,68)(49,54)(51,67)(57,65)(58,66);(2,4)(5,8,12,31)(6,14,18,46)(7,20,24,57)(9,30,33,65)(10,35,36,29)(11,28,32,66)(13,38,41,68)(15,48,37,45)(16,49,34,26)(17,51,52,44)(19,53,50,43)(21,27,59,64)(22,58,25,56)(23,55,60,63)(39,62,42,67)(40,54,47,61)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fh,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, 55147);