#8228389

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

4189 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,38,40)(35,39,43,48)(36,41,45,46)(37,44,47,42);(2,7)(4,8)(9,14)(12,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,35)(34,39)(36,44)(37,46)(38,43)(40,48)(41,47)(42,45);(5,13)(9,14)(10,16)(12,15)(18,21)(23,25)(24,28)(31,32)(34,40)(35,43)(41,46)(42,44);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,36)(34,41)(35,42)(37,39)(38,45)(40,46)(43,44)(47,48);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,37)(34,42)(35,41)(36,39)(38,47)(40,44)(43,46)(45,48);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,48)(44,45)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8135585);