#8934260

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

4103 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841456);