#8634175

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

4121 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541371);