#8901456

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

4167 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(33,34,38,42)(35,43,44,39)(36,40,45,48)(37,41,46,47);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(35,44)(37,46)(39,43)(41,47);(2,6)(8,9)(10,13)(15,16)(18,22)(23,29)(24,30)(31,32)(33,35,38,44)(34,39,42,43)(36,37,45,46)(40,47,48,41);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31)(33,36)(34,40)(35,37)(38,45)(39,47)(41,43)(42,48)(44,46);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)(33,37)(34,41)(35,45)(36,44)(38,46)(39,40)(42,47)(43,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8808652);