#419892

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327088);