#501460

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 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)(7,12)(8,10)(9,11)(13,14,19,31,29,38,30,21,16,22,17,23,18,24,33,37)(15,25,32,35,42,36,43,39,26,20,27,40,28,41,44,34);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,26)(14,20,22,25)(17,27,19,32)(18,28,29,42)(21,34,37,39)(23,35,31,40)(24,36,38,41)(30,43,33,44);(2,4)(5,8,9,7)(6,10,11,12)(14,21)(17,30)(18,29)(19,33)(20,34)(22,37)(23,24)(25,39)(27,43)(28,42)(31,38)(32,44)(35,36)(40,41);(6,11)(10,12)(14,22)(15,26)(21,37)(23,31)(24,38)(27,32)(28,42)(43,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=ei,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, 408656);