#147928

order	512
nilpotency class	4
isoclinism	isoclinic class of #140781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
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

950 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)(7,18)(8,23,26,13)(9,28)(10,34)(11,15)(12,20)(14,32)(16,47)(17,31)(19,54,56,53)(21,48)(22,39)(24,62,63,68)(25,49,55,46)(27,44,36,52)(29,51)(30,43,66,42)(33,50)(35,45)(37,58,67,64)(38,59)(40,65)(41,60,61,57);(2,4)(5,7,11,22)(6,12,15,38)(8,24,26,63)(9,29,33,65)(10,35,31,21)(13,41,23,61)(14,45,16,48)(17,51,34,40)(18,50,39,28)(19,55,56,25)(20,47,59,32)(27,64,36,58)(30,60,66,57)(37,42,67,43)(44,53,52,54)(46,68,49,62);(5,8)(6,13)(7,19)(9,30)(10,36)(11,26)(12,37)(14,46)(15,23)(16,49)(17,52)(18,53)(20,58)(21,60)(22,56)(24,29)(25,32)(27,31)(28,42)(33,66)(34,44)(35,57)(38,67)(39,54)(40,62)(41,48)(43,50)(45,61)(47,55)(51,68)(59,64)(63,65)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,f^2=hi,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=dh,f^b=fh,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, 55124);