#9224370

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

4024 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)(3,8)(4,14)(5,6)(7,10)(9,13)(11,16)(12,15)(17,18)(19,29)(20,26)(21,22)(23,27)(24,28)(25,32)(30,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,8)(4,15)(5,11)(6,16)(7,13)(9,10)(12,14)(17,19,24,32)(18,25,28,29)(20,30,21,27)(22,31,26,23)(36,44)(39,46)(42,48)(45,47);(1,4,7,5)(2,6,10,14)(3,11,13,15)(8,12,9,16)(17,20,24,21)(18,26,28,22)(19,30,32,27)(23,29,31,25)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,4)(2,6,10,14)(3,11,13,15)(8,16,9,12)(17,21,24,20)(18,26,28,22)(19,27,32,30)(23,29,31,25);(1,6)(2,5)(3,12)(4,10)(7,14)(8,15)(9,11)(13,16)(17,22,24,26)(18,21,28,20)(19,31,32,23)(25,27,29,30)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,3)(2,9)(4,11)(5,15)(6,16)(7,13)(8,10)(12,14)(17,23,24,31)(18,27,28,30)(19,22,32,26)(20,29,21,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gi,e^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, 9131566);