#494021

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#20723
prime radical	#3455	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,g^b=gi,d^c=di,g^c=gh,g^f=gi

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, 401217);