#8402464

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

3975 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)(3,5)(4,9)(6,11)(7,8)(10,15)(12,14)(13,16)(17,18,23,20)(19,24,27,29)(21,25,32,30)(22,26,28,31)(33,34,39,41)(35,37,43,36)(38,40,46,48)(42,45,47,44);(1,3,7,10)(2,5,8,15)(4,12,13,6)(9,14,16,11)(17,19)(18,24)(20,29)(21,22)(23,27)(25,26)(28,32)(30,31)(33,35,39,43)(34,36,41,37)(38,42,46,47)(40,44,48,45);(2,8)(3,10)(6,12)(9,16)(19,27)(22,28)(24,29)(26,31)(33,36,39,37)(34,35,41,43)(38,45,46,44)(40,47,48,42);(1,4)(2,9)(3,12)(5,14)(6,10)(7,13)(8,16)(11,15)(17,20,23,18)(19,24,27,29)(21,25,32,30)(22,31,28,26)(33,37,39,36)(34,35,41,43)(38,45,46,44)(40,42,48,47);(1,5,7,15)(2,10,8,3)(4,11,13,14)(6,9,12,16)(17,21)(18,25)(19,28)(20,30)(22,27)(23,32)(24,31)(26,29)(33,35)(34,37)(36,41)(38,42)(39,43)(40,45)(44,48)(46,47);(1,6,7,12)(2,11,8,14)(3,4,10,13)(5,9,15,16)(17,22,23,28)(18,26,20,31)(19,21,27,32)(24,25,29,30)(33,38,39,46)(34,40,41,48)(35,42,43,47)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309660);