#9057377

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	#58892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8964573);