#8159274

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

3863 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(2,3)(7,9)(8,10)(14,15)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(33,34,38,40)(35,36,43,41)(37,39,46,45)(42,44,48,47);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,41)(34,35,40,43)(37,44,46,47)(39,42,45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=g,e^2=gh,f^2=h,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, 8066470);