#146911

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1824
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 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)(7,17,20,12)(8,22)(9,26)(10,32)(11,14)(13,24)(15,30)(16,29)(18,45,46,64)(19,49,27,59)(21,36,51,44)(23,54)(25,57)(28,42)(31,58)(33,41)(34,47,43,56)(35,52,40,61)(37,62)(38,55)(39,63)(48,66,53,68)(50,65,60,67);(1,2,3,4)(5,7)(6,12)(8,18)(9,27)(10,21)(11,20)(13,34)(14,17)(15,40)(16,36)(19,31)(22,45)(23,46)(24,56)(25,48)(26,49)(28,60)(29,51)(30,61)(32,44)(33,53)(35,39)(37,65)(38,43)(41,68)(42,67)(47,55)(50,62)(52,63)(54,64)(57,66)(58,59);(5,8,10,33,11,23,29,25)(6,13,32,37,14,38,16,42)(7,18,21,53,20,46,51,48)(9,28,30,24,31,62,63,55)(12,34,44,65,17,43,36,67)(15,41,58,54,39,57,26,22)(19,50,52,47,27,60,61,56)(35,66,49,45,40,68,59,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,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, 54107);