#501456

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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	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

544 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)(13,14,17,22,29,37,43,35,16,21,28,36,18,23,30,20)(15,24,26,39,41,34,44,32,25,19,31,33,27,40,42,38);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,25)(14,19,21,24)(17,26,28,31)(18,27,29,41)(20,32,35,38)(22,33,36,39)(23,34,37,40)(30,42,43,44);(2,4)(5,8,9,7)(6,10,11,12)(14,20)(17,30)(18,29)(19,32)(21,35)(22,23)(24,38)(26,42)(27,41)(28,43)(31,44)(33,34)(36,37)(39,40);(6,11)(10,12)(14,21)(15,25)(20,35)(22,36)(23,37)(26,31)(27,41)(42,44)

Polycyclic presentation:

2^9|a^2=f,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, 408652);