#8522184

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

4107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,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, 8429380);