#9863927

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3827 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)(3,7)(4,8)(5,13)(6,10)(9,15)(11,12)(14,16)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48);(1,3,6,8)(2,7,10,4)(5,11,15,16)(9,14,13,12)(17,18,23,20)(19,26,24,21)(22,25,31,28)(27,30,32,29)(33,34,39,44)(35,40,47,48)(36,41,45,46)(37,38,43,42);(1,2,6,10)(3,4,8,7)(5,9,15,13)(11,12,16,14)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35,39,47)(34,40,44,48)(36,37,45,43)(38,46,42,41);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,16)(13,15)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,36,39,45)(34,41,44,46)(35,37,47,43)(38,48,42,40);(1,4,6,7)(2,8,10,3)(5,14,15,12)(9,11,13,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,37)(34,42)(35,45)(36,47)(38,44)(39,43)(40,41)(46,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,14)(8,16)(10,13)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,48,45,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=ghi,e^2=i,f^2=g,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, 9771123);