#8114376

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

3979 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)(19,27)(20,30)(21,23)(22,32)(24,29)(25,31)(26,28)(34,36)(37,45)(39,42)(43,47);(2,3)(7,9)(8,10)(14,15)(17,19)(18,24)(20,22)(21,27)(23,29)(25,32)(26,30)(28,31)(34,36)(35,41)(40,44)(43,47);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021572);