#8334331

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gi,e^2=g,f^2=hi,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, 8241527);