#148200

order	512
nilpotency class	4
isoclinism	isoclinic class of #137655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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)(5,6,11,18,12,19,35,16,13,20,34,48,10,17,9,33)(7,14,24,40,23,39,56,38,25,41,58,65,36,49,22,55)(8,26,30,46,31,47,62,43,32,50,61,15,29,44,28,45)(21,51,53,64,27,59,60,66,54,67,68,37,57,63,52,42);(1,2,3,4)(5,7)(6,14)(8,27)(9,22)(10,36)(11,24)(12,23)(13,25)(15,42)(16,38)(17,49)(18,40)(19,39)(20,41)(21,29)(26,59)(28,53)(30,60)(31,54)(32,57)(33,55)(34,58)(35,56)(37,43)(44,51)(45,64)(46,66)(47,67)(48,65)(50,63)(52,61)(62,68);(5,8,13,32)(6,15,20,46)(7,21,25,54)(9,30,35,61)(10,31,12,29)(11,28,34,62)(14,37,41,64)(16,47,33,44)(17,45,19,43)(18,50,48,26)(22,53,56,68)(23,57,36,27)(24,52,58,60)(38,59,55,63)(39,66,49,42)(40,67,65,51)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,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, 55396);