#8688411

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

3949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595607);