#503546

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=efh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di

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, 410742);