#8932126

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

4020 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:

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

Polycyclic presentation:

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

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, 8839322);