#147933

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
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)(7,17)(8,22,26,13)(9,28)(10,34)(11,15)(12,19)(14,32)(16,31)(18,52,54,50)(20,59)(21,51)(23,62,53,68)(24,45,25,64)(27,43,36,48)(29,47)(30,41,63,42)(33,49)(35,44)(37,57,67,65)(38,58)(39,66)(40,55,61,60)(46,56);(2,4)(5,7,11,21)(6,12,15,38)(8,23,26,53)(9,29,33,66)(10,35,31,20)(13,40,22,61)(14,44,46,59)(16,47,34,39)(17,49,51,28)(18,24,54,25)(19,56,58,32)(27,65,36,57)(30,60,63,55)(37,41,67,42)(43,52,48,50)(45,62,64,68);(5,8)(6,13)(7,18)(9,30)(10,36)(11,26)(12,37)(14,45)(15,22)(16,48)(17,50)(19,57)(20,60)(21,54)(23,29)(24,56)(25,32)(27,31)(28,42)(33,63)(34,43)(35,55)(38,67)(39,62)(40,44)(41,49)(46,64)(47,68)(51,52)(53,66)(58,65)(59,61)

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=fh,h^a=hi,c^b=cf,d^b=dh,f^b=fh,h^b=hi,d^c=dhi,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, 55129);