#8688388

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

Polycyclic presentation:

2^9|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, 8595584);