#498021

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

567 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)(11,15)(12,14)(13,16)(17,18,21,25,22,26,32,38,20,23,30,34,31,35,42,45)(19,24,28,36,29,37,41,46,27,33,39,43,40,44,47,48);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(18,23)(24,33)(25,34)(26,35)(36,43)(37,44)(38,45)(46,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,19,20,27)(18,24,23,33)(21,28,30,39)(22,29,31,40)(25,36,34,43)(26,37,35,44)(32,41,42,47)(38,46,45,48);(2,5)(6,8)(11,16)(13,15)(19,27)(24,33)(28,39)(29,40)(36,43)(37,44)(41,47)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=dei,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, 405217);