#506455

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

581 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:

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

Polycyclic presentation:

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

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, 413651);