#501905

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ei,d^2=i,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, 409101);