#505641

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,31)(21,26)(22,35)(23,24)(25,33)(27,37)(29,45)(30,42)(32,47)(34,40)(36,44)(38,43)(39,48)(41,46);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,23)(18,24,26,28)(20,29,32,38)(22,30,34,37)(25,39,41,44)(27,40,42,35)(31,43,47,45)(33,36,46,48);(11,15)(13,16)(17,20,21,32)(18,25,26,41)(19,29,23,38)(22,36,34,48)(24,39,28,44)(27,43,42,45)(30,46,37,33)(31,40,47,35);(1,3,4,8)(2,6,5,7)(18,26)(24,28)(25,41)(27,42)(31,47)(35,40)(39,44)(43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412837);