#505254

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

467 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,43)(30,38)(31,39)(32,40)(33,42)(34,41)(35,44)(36,45)(37,47)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,29)(23,36)(25,39)(26,38)(27,43)(28,45)(32,35)(33,46)(34,47)(37,41)(40,44)(42,48);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(18,25)(19,29)(20,23)(22,35)(24,37)(27,42)(30,32)(31,47)(33,34)(36,46)(38,48)(39,43)(40,45)(41,44);(2,5)(6,8)(9,12,13,11)(10,16,15,14)(17,20,21,23)(18,26,25,28)(19,30,31,36)(22,33,34,35)(24,38,39,45)(27,42,41,44)(29,46,47,32)(37,40,43,48)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=e,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,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, 412450);