#8745910

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

3947 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=ghi,e^2=g,f^2=gh,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, 8653106);