#509030

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

488 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416226);