#10247497

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	#49887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

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

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, 10154693);