#8932274

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

4131 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)(3,8)(4,9)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34,39,42)(35,40,43,47)(36,46,45,38)(37,41,44,48);(2,7)(4,14)(8,16)(10,12)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29)(33,35,39,43)(34,40,42,47)(36,44,45,37)(38,41,46,48);(3,11)(5,15)(8,16)(10,12)(18,25)(19,20)(22,31)(29,32)(35,43)(36,45)(38,46)(40,47);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,20,23,19)(18,24,25,21)(22,28,31,27)(26,32,30,29)(33,36)(34,38)(35,44)(37,43)(39,45)(40,41)(42,46)(47,48);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,21,23,24)(18,20,25,19)(22,29,31,32)(26,28,30,27)(33,37,39,44)(34,41,42,48)(35,45,43,36)(38,47,46,40);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,38)(34,36)(35,41)(37,47)(39,46)(40,44)(42,45)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=gh,e^2=gi,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, 8839470);