#8633739

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	#41041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(33,34,38,41)(35,37,40,47)(36,43,46,39)(42,48,44,45);(2,4)(3,10)(8,12)(11,13)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(2,4)(7,9)(8,12)(15,16)(18,23)(20,29)(24,31)(26,32)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,35)(34,37)(36,42)(38,40)(39,45)(41,47)(43,48)(44,46);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,43)(47,48);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37)(34,40)(35,41)(36,45)(38,47)(39,44)(42,43)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540935);