#10069105

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	#52328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(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)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(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)(20,31)(23,27)(26,28)(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)(18,25)(19,28)(21,27)(22,31)(23,32)(24,29)(26,30)(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)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(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)(17,21)(18,26)(19,29)(20,27)(22,32)(23,31)(24,28)(25,30)(33,38)(34,42)(35,43)(36,41)(37,40)(39,47)(44,45)(46,48)

Polycyclic presentation:

2^9|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, 9976301);