#8159819

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	#58030	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

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

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, 8067015);