#8749667

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8656863);