#9924021

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	#55441	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,22,21)(18,23,25,26)(20,27,29,28)(24,30,32,31)(35,43)(37,46)(39,48)(41,47);(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,35,38,43)(34,39,42,48)(36,37,44,46)(40,47,45,41);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,36,38,44)(34,40,42,45)(35,37,43,46)(39,47,48,41);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,32,31)(33,37)(34,41)(35,44)(36,43)(38,46)(39,40)(42,47)(45,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,46)(43,48)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gi,e^2=h,f^2=gh,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, 9831217);