#508796

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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

509 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415992);