#10216520

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10123716);