#147931

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
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

946 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)(5,6,11,18,12,16,37,19)(7,20,24,42,25,51,38,52)(8,26,30,49,31,14,39,48)(9,32,33,15,34,17,10,35)(13,23,43,22,41,58,44,59)(21,53,56,68,57,45,61,64)(27,40,60,66,63,65,55,67)(28,62,29,50,36,47,54,46);(5,7,12,25)(6,13,16,41)(8,27,31,63)(9,23,34,58)(10,22,33,59)(11,38,37,24)(14,45,26,53)(15,42,35,52)(17,20,32,51)(18,44,19,43)(21,54,57,29)(28,56,36,61)(30,55,39,60)(40,62,65,47)(46,67,50,66)(48,68,49,64);(2,4)(5,8)(6,14)(7,21)(9,29)(10,36)(11,39)(12,31)(13,40)(15,47)(16,26)(17,50)(18,49)(19,48)(20,45)(22,27)(23,60)(24,61)(25,57)(28,33)(30,37)(32,46)(34,54)(35,62)(38,56)(41,65)(42,68)(43,67)(44,66)(51,53)(52,64)(55,58)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dh,f^c=fh,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, 55127);