#10171713

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	#49114	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=hi,e^2=gi,f^2=g,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, 10078909);