#9863167

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3822 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770363);