#417685

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2301 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:

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

Polycyclic presentation:

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

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, 324881);