#10069106

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	#52323	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=ghi,e^2=i,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, 9976302);