#9924020

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

3877 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 9831216);