#9039162

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	#52048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(3,10)(4,7)(5,13)(6,8)(9,14)(11,15)(12,16)(19,20)(22,24)(27,28)(29,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,6,4)(2,7,8,10)(5,11,14,12)(9,15,13,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,29)(28,31)(30,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,8)(5,14)(7,10)(11,12)(17,19,23,20)(18,24,26,22)(21,28,30,27)(25,29,32,31)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,4,6,3)(2,7,8,10)(5,12,14,11)(9,15,13,16)(17,20,23,19)(18,24,26,22)(21,27,30,28)(25,29,32,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,2)(3,7)(4,10)(5,9)(6,8)(11,15)(12,16)(13,14)(17,21,23,30)(18,25,26,32)(19,27,20,28)(22,29,24,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,5,6,14)(2,9,8,13)(3,11,4,12)(7,15,10,16)(17,22,23,24)(18,19,26,20)(21,29,30,31)(25,27,32,28)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946358);