#8522181

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

3965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8429377);