#8963278

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

3846 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)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,18,23,21)(19,25,20,24)(22,29,30,26)(27,31,28,32)(35,37)(39,41)(43,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(18,21)(24,25)(26,29)(31,32)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(3,9)(5,13)(7,10)(8,12)(17,19,23,20)(18,24,21,25)(22,28,30,27)(26,32,29,31)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870474);