#10040720

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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)(5,12,13,8)(11,16,15,14)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,34,39,41)(35,40,38,48)(36,44,46,37)(42,47,45,43);(2,4)(3,9)(8,12)(11,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,38)(34,40,41,48)(36,45,46,42)(37,47,44,43);(2,4)(3,9)(8,12)(11,15)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,36,39,46)(34,37,41,44)(35,42,38,45)(40,43,48,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,44)(47,48);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,37)(34,36)(35,43)(38,47)(39,44)(40,42)(41,46)(45,48);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,35)(34,40,41,48)(36,45,46,42)(37,43,44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947916);