#501518

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

615 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)(7,11)(8,9)(10,12)(13,14,19,25,17,31,32,36,18,24,33,43,20,23,35,40)(15,26,28,42,29,39,34,22,16,21,30,38,27,37,44,41);(1,2)(3,4)(6,9)(10,11)(13,15)(14,21)(16,18)(17,29)(19,28)(20,27)(22,40)(23,39)(24,26)(25,38)(30,33)(31,37)(32,34)(35,44)(36,41)(42,43);(2,4)(13,16,18,15)(14,22,24,41)(17,29,20,27)(19,34,33,44)(21,36,26,40)(23,38,31,42)(25,39,43,37)(28,35,30,32);(5,7,8,12)(6,10,9,11)(13,17,18,20)(14,23,24,31)(15,27,16,29)(19,32,33,35)(21,37,26,39)(22,38,41,42)(25,40,43,36)(28,44,30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408714);