#9062131

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969327);