#9236444

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(4,15)(5,10)(6,14)(7,11)(9,13)(12,16)(17,18,24,22)(19,28,21,25)(20,26,31,29)(23,27,32,30)(33,34,39,43)(35,37,41,48)(36,42,38,40)(44,47,45,46);(1,3)(2,8)(4,5)(6,11)(7,14)(9,16)(10,15)(12,13)(17,19,24,21)(18,25,22,28)(20,27,31,30)(23,29,32,26)(33,35)(34,37)(36,44)(38,45)(39,41)(40,46)(42,47)(43,48);(1,4)(2,9)(3,12)(5,14)(6,16)(7,13)(8,10)(11,15)(17,20,24,31)(18,26,22,29)(19,27,21,30)(23,28,32,25)(33,36,39,38)(34,40,43,42)(35,44,41,45)(37,46,48,47);(1,4)(2,9)(3,5)(6,10)(7,13)(8,16)(11,15)(12,14)(17,21,24,19)(18,25,22,28)(20,30,31,27)(23,29,32,26);(1,5)(2,10)(3,13)(4,14)(6,15)(7,12)(8,9)(11,16)(17,22,24,18)(19,28,21,25)(20,29,31,26)(23,27,32,30)(33,37)(34,41)(35,43)(36,46)(38,47)(39,48)(40,45)(42,44);(1,6,7,8)(2,3,11,14)(4,16,13,10)(5,9,12,15)(17,23)(18,27)(19,29)(20,28)(21,26)(22,30)(24,32)(25,31)(33,38,39,36)(34,42,43,40)(35,45,41,44)(37,47,48,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143640);