#9814851

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
prime radical	#50479	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(19,25)(23,26)(27,29)(31,32)(33,34,39,44)(35,40,47,48)(36,41,45,46)(37,38,43,42);(2,9)(3,12)(4,14)(10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,35,39,47)(34,40,44,48)(36,37,45,43)(38,46,42,41);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,36,39,45)(34,41,44,46)(35,37,47,43)(38,48,42,40);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,37)(34,42)(35,45)(36,47)(38,44)(39,43)(40,41)(46,48);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,48,45,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9722047);