#509725

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416921);