#427892

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1191 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 335088);