#148643

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(7,19,23,13)(8,24)(9,30)(10,18)(11,17)(12,15)(14,27)(16,33)(20,49,50,65)(21,52,55,62)(22,38,35,46)(25,57,56,68)(26,59)(28,44)(29,58)(31,40)(32,43)(34,45)(36,51,47,63)(37,54,48,64)(39,67,66,53)(41,61)(42,60);(1,2)(3,4)(5,7)(6,13)(8,25)(9,21)(10,35)(11,22)(12,23)(14,39)(15,19)(16,37)(17,46)(18,38)(20,28)(24,57)(26,56)(27,53)(29,50)(30,62)(31,63)(32,64)(33,54)(34,55)(36,40)(41,47)(42,66)(43,48)(44,49)(45,52)(51,61)(58,65)(59,68)(60,67);(2,4)(5,8,11,28,12,26,10,29)(6,14,17,41,15,42,18,40)(7,20,22,56,23,50,35,25)(9,31,33,27,34,61,32,60)(13,36,46,39,19,47,38,66)(16,44,45,59,43,58,30,24)(21,53,54,51,55,67,64,63)(37,57,52,49,48,68,62,65)

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,h^c=hi

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, 55839);