#394639

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2100 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:

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

Polycyclic presentation:

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

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, 301835);