#9164016

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,21)(19,24,27,29)(20,25,31,30)(22,26,32,28)(33,34)(35,44)(36,41)(37,38)(39,43)(40,47)(42,46)(45,48);(2,7)(4,15)(8,11)(10,13)(17,19,23,27)(18,24,21,29)(20,28,31,26)(22,30,32,25)(33,35)(34,40)(36,45)(37,46)(38,41)(39,47)(42,48)(43,44);(2,7)(3,12)(9,14)(10,13)(19,27)(20,31)(24,29)(25,30)(33,36,39,46)(34,41,43,42)(35,45,47,37)(38,44,48,40);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,25)(19,28)(21,30)(22,24)(23,31)(26,27)(29,32)(33,36)(34,42)(35,45)(37,47)(38,44)(39,46)(40,48)(41,43);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,18)(19,29,27,24)(20,25,31,30)(22,26,32,28)(33,37,39,45)(34,38,43,48)(35,46,47,36)(40,41,44,42);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22)(18,26)(19,30)(20,24)(21,28)(23,32)(25,27)(29,31)(33,38)(34,37)(35,41)(36,44)(39,48)(40,46)(42,47)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071212);