#10159324

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

3845 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,e^2=hi,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, 10066520);