#147263

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1862
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54459);