#10040736

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3737 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)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,34)(35,40)(36,44)(37,42)(38,46)(39,43)(41,47)(45,48);(2,4)(3,9)(8,12)(11,15)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(33,35,39,38)(34,40,43,46)(36,45,47,42)(37,44,48,41);(2,4)(3,9)(8,12)(11,15)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,36,39,47)(34,41,43,44)(35,42,38,45)(37,40,48,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,20)(18,26,27,25)(19,29,22,30)(24,31,28,32)(33,36)(34,41)(35,42)(37,46)(38,45)(39,47)(40,48)(43,44);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,22,23,19)(18,24,27,28)(20,29,21,30)(25,32,26,31)(33,37)(34,42)(35,41)(36,46)(38,44)(39,48)(40,47)(43,45);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,31)(30,32)(33,38,39,35)(34,40,43,46)(36,45,47,42)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=hi,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, 9947932);