#9924009

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	#48561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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:

(1,2,7,4)(3,10,6,8)(5,9,13,12)(11,16,14,15)(17,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,32)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,30,32,29)(35,43)(37,46)(39,48)(41,47);(2,4)(8,10)(9,12)(15,16)(17,19,23,20)(18,21,26,24)(22,27,31,28)(25,29,32,30)(33,35,38,43)(34,39,42,48)(36,37,44,46)(40,47,45,41);(1,4,7,2)(3,10,6,8)(5,12,13,9)(11,16,14,15)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,30,32,29)(33,36,38,44)(34,40,42,45)(35,37,43,46)(39,47,48,41);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,24)(18,19,26,20)(22,30,31,29)(25,28,32,27)(33,37)(34,41)(35,44)(36,43)(38,46)(39,40)(42,47)(45,48);(1,6,7,3)(2,10,4,8)(5,11,13,14)(9,15,12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(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=ghi,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, 9831205);