#8751690

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

3737 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=ghi,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, 8658886);