#8749819

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3825 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)(10,14)(11,16)(13,15)(17,18)(19,27)(20,21)(22,24)(23,26)(25,31)(28,29)(30,32)(33,34)(35,41)(36,38)(37,44)(39,42)(40,45)(43,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,31,32)(34,36)(35,42)(40,44)(43,48);(2,3)(7,9)(8,10)(15,16)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,31,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,25)(18,21,26,31)(19,28,22,30)(24,32,27,29)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,47,48,46);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,25)(19,29)(20,26)(22,32)(23,31)(24,28)(27,30)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,19)(18,24,26,27)(20,28,25,30)(21,29,31,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=h,e^2=gh,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, 8657015);