#9222124

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	#59952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9129320);