#9009218

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8916414);