#495436

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
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, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,20,28,21,29,30,41,22,27,31,39,32,40,42,47)(18,23,25,34,26,35,38,45,24,33,36,43,37,44,46,48);(9,12,13,11)(10,14,15,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48);(1,3,4,7)(2,6,5,8)(18,24)(23,33)(25,36)(26,37)(34,43)(35,44)(38,46)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci

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, 402632);