#8824402

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731598);