#9685570

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	#47228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,24,28)(19,27,23,25)(20,21,22,26)(29,32,31,30)(33,34)(35,42)(36,40)(37,47)(38,43)(39,44)(41,45)(46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,24,23)(18,25,28,27)(20,29,22,31)(21,30,26,32)(33,35)(34,38)(36,45)(37,46)(39,43)(40,47)(41,48)(42,44);(2,3)(7,9)(8,10)(14,15)(17,20,24,22)(18,26,28,21)(19,29,23,31)(25,32,27,30)(33,36,39,46)(34,40,44,48)(35,45,43,37)(38,47,42,41);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,24,26)(18,20,28,22)(19,30,23,32)(25,29,27,31)(33,37,39,45)(34,41,44,47)(35,46,43,36)(38,48,42,40);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22,24,20)(18,26,28,21)(19,29,23,31)(25,30,27,32)(33,35)(34,42)(36,45)(37,46)(38,44)(39,43)(40,41)(47,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,23,24,19)(18,27,28,25)(20,31,22,29)(21,32,26,30)(33,38,39,42)(34,43,44,35)(36,47,46,41)(37,48,45,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9592766);