#8747890

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655086);