#8115399

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022595);