#9214724

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,f^2=g,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, 9121920);