#10171864

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	#40297	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18,22,25)(19,24,28,23)(20,21,31,29)(26,32,30,27)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,42,44,48);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,18,22,25)(19,24,28,23)(20,29,31,21)(26,27,30,32)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,48,46,42);(2,4)(8,10)(9,12)(15,16)(17,19,22,28)(18,23,25,24)(20,26,31,30)(21,27,29,32)(33,34,39,37)(35,40,45,43)(36,41,47,46)(38,42,44,48);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,25)(26,31)(29,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,48)(42,43);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,20,22,31)(18,21,25,29)(19,26,28,30)(23,27,24,32)(33,37,39,34)(35,43,45,40)(36,41,47,46)(38,42,44,48);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,21)(18,20)(19,27)(22,29)(23,26)(24,30)(25,31)(28,32)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,41,43,46)

Polycyclic presentation:

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

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, 10079060);