#8718957

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4152 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8626153);