#8750809

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

3725 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)(4,10)(5,8)(6,12)(7,9)(13,16)(14,15)(17,18,23,22)(19,26,20,24)(21,25,29,30)(27,32,28,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3,7,5)(2,8,9,11)(4,12,14,13)(6,10,16,15)(17,19,23,20)(18,24,22,26)(21,27,29,28)(25,31,30,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(2,9)(4,14)(8,11)(12,13)(17,18,23,22)(19,24,20,26)(21,25,29,30)(27,31,28,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,20,23,19)(18,24,22,26)(21,28,29,27)(25,31,30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5,7,3)(2,11,9,8)(4,12,14,13)(6,10,16,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,6,7,16)(2,12,9,13)(3,10,5,15)(4,8,14,11)(17,22,23,18)(19,26,20,24)(21,25,29,30)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=h,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, 8658005);