#420394

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1293 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=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, 327590);