#505362

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

559 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,25)(21,27)(22,28)(23,26)(24,32)(30,41)(31,40)(33,39)(34,43)(35,45)(36,48)(37,44)(38,42)(46,47);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,21,32)(18,24,27,29)(20,31,23,38)(22,36,35,47)(25,42,26,40)(28,46,45,48)(30,37,33,34)(39,44,41,43);(10,14)(13,15)(17,20)(18,25)(19,30)(21,23)(24,39)(26,27)(29,41)(31,36)(32,33)(34,37)(38,47)(40,48)(42,46)(43,44);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,20,21,23)(18,26,27,25)(19,31,32,38)(22,34,35,37)(24,40,29,42)(28,44,45,43)(30,47,33,36)(39,46,41,48)

Polycyclic presentation:

2^9|b^2=fi,d^2=ef,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, 412558);