#9923902

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

4027 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(18,21)(24,25)(26,29)(31,32)(35,43)(37,46)(39,48)(41,47);(2,7)(3,10)(8,13)(11,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,38,43)(34,39,42,48)(36,37,44,46)(40,47,45,41);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,38,44)(34,40,42,45)(35,37,43,46)(39,47,48,41);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,37)(34,41)(35,44)(36,43)(38,46)(39,40)(42,47)(45,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,46)(43,48)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831098);