#148530

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 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:

(5,6)(7,20)(8,27,32,48)(9,17)(10,15)(11,37)(12,19)(13,22)(14,30,45,29)(16,36)(18,34)(21,53,55,68)(23,42)(24,50)(25,47)(26,52)(28,43,60,67)(31,59,63,46)(33,64,61,44)(35,49)(38,40)(39,62,66,56)(41,58)(51,54,65,57);(1,2)(3,4)(5,7)(6,13)(8,28)(9,24)(10,22)(11,38)(12,26)(14,43)(15,40)(16,47)(17,20)(18,50)(19,42)(21,31)(23,36)(25,34)(27,53)(29,56)(30,62)(32,60)(33,54)(35,58)(37,41)(39,44)(45,67)(46,51)(48,68)(49,52)(55,63)(57,61)(59,65)(64,66);(2,4)(5,8,9,29,34,61,36,63,12,32,35,30,11,33,10,31)(6,14,15,27,37,59,49,64,19,45,16,48,18,46,17,44)(7,21,22,54,38,62,58,60,26,55,23,57,25,56,24,28)(13,39,20,51,50,68,47,67,42,66,52,65,41,53,40,43)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55726);