#420844

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1819 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)(5,6,12,18)(7,19,23,39)(8,25,30,42)(9,31,35,45)(10,17,11,16)(13,24,38,22)(14,28,43,29)(15,33,46,34)(20,36,26,40)(21,52,53,37)(27,57,61,41)(32,44,54,48)(47,50,65,51)(49,68,56,67)(55,60,62,59)(58,66,63,64);(5,7,10,22,12,23,11,24)(6,13,16,19,18,38,17,39)(8,26,29,50,30,20,28,51)(9,32,33,53,35,54,34,21)(14,40,25,47,43,36,42,65)(15,44,45,52,46,48,31,37)(27,58,60,56,61,63,59,49)(41,67,62,66,57,68,55,64);(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,30)(13,36)(15,41)(16,42)(17,25)(18,43)(19,47)(21,49)(22,50)(23,26)(24,51)(31,62)(32,63)(33,59)(34,60)(35,61)(37,64)(38,40)(39,65)(44,68)(45,55)(46,57)(48,67)(52,66)(53,56)(54,58);(2,4)(5,9,12,35)(6,15,18,46)(7,21,23,53)(8,27,30,61)(10,34,11,33)(13,37,38,52)(14,41,43,57)(16,31,17,45)(19,48,39,44)(20,49,26,56)(22,54,24,32)(25,55,42,62)(28,60,29,59)(36,64,40,66)(47,67,65,68)(50,58,51,63)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,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,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 328040);