#9171630

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	#55328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18)(19,26)(20,21)(22,24)(23,25)(27,28)(29,32)(30,31)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19,23,22)(18,24,25,26)(20,27,29,30)(21,31,32,28)(33,35)(34,36)(37,39)(38,43)(40,42)(41,44)(45,47)(46,48);(3,11)(7,8)(12,14)(13,16)(18,25)(20,29)(24,26)(27,30)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,23,29)(18,21,25,32)(19,27,22,30)(24,31,26,28)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,46)(47,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,21)(18,20)(19,28)(22,31)(23,32)(24,30)(25,29)(26,27)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,46,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,22,23,19)(18,26,25,24)(20,30,29,27)(21,28,32,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=gh,f^2=g,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, 9078826);