#494098

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

670 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:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(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);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,33)(25,34)(26,35)(30,41)(31,39)(32,40)(36,43)(37,44)(38,45)(42,47)(46,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48);(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=ef,c^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, 401294);