#497252

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

406 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27,23,19,28,31,21,26,36,42,37,30,24,38)(20,32,34,46,35,44,43,48,33,25,40,41,47,29,39,45);(2,5)(3,8)(17,19,22,31,23,26,28,42,21,30,36,38,37,18,24,27)(20,29,34,45,35,32,43,46,33,44,40,48,47,25,39,41);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,21,33)(18,25,26,32)(19,29,30,44)(22,34,36,40)(23,35,37,47)(24,39,28,43)(27,41,42,46)(31,45,38,48);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(20,33)(27,42)(31,38)(34,40)(35,47)(39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404448);