#147141

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54337);