#507523

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=eg,c^2=ef,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414719);