#10069249

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	#44342	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9976445);