#9210287

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3881 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)(3,8)(4,12)(5,7)(6,11)(9,14)(10,15)(13,16)(17,18,23,22)(19,24,27,28)(20,29,30,25)(21,26,31,32)(33,34,39,37)(35,40,45,46)(36,42,38,41)(43,48,44,47);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,19)(18,24)(20,21)(22,28)(23,27)(25,32)(26,29)(30,31)(33,35)(34,40)(36,43)(37,46)(38,44)(39,45)(41,47)(42,48);(2,7)(6,8)(9,12)(15,16)(19,27)(20,30)(24,28)(25,29)(33,36,39,38)(34,41,37,42)(35,43,45,44)(40,47,46,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,20,23,30)(18,25,22,29)(19,21,27,31)(24,32,28,26);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,21)(18,26)(19,20)(22,32)(23,31)(24,29)(25,28)(27,30)(33,37,39,34)(35,40,45,46)(36,42,38,41)(43,47,44,48);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,22,23,18)(19,28,27,24)(20,29,30,25)(21,26,31,32)(33,38,39,36)(34,42,37,41)(35,44,45,43)(40,48,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9117483);