#148051

order	512
nilpotency class	4
isoclinism	isoclinic class of #136643
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=ehi,f^c=fi,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, 55247);