#8291607

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

4067 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:

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

Polycyclic presentation:

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

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, 8198803);