#510047

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,9,11)(7,8,12,10)(13,14,19,31,29,39,30,21,16,22,17,23,18,24,33,38)(15,25,32,36,42,37,43,35,26,20,27,40,28,41,44,34);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15)(14,20)(16,26)(17,27)(18,28)(19,32)(21,34)(22,25)(23,36)(24,37)(29,42)(30,43)(31,40)(33,44)(35,38)(39,41);(2,4)(14,21)(17,30)(18,29)(19,33)(20,34)(22,38)(23,24)(25,35)(27,43)(28,42)(31,39)(32,44)(36,37)(40,41);(5,8,9,10)(6,7,11,12)(14,21)(15,26)(17,30)(18,29)(19,33)(20,35)(22,38)(23,24)(25,34)(27,44)(31,39)(32,43)(36,41)(37,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,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, 417243);