#8745898

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

4100 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)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(1,2,5,6)(3,7,11,9)(4,8,14,12)(10,15,13,16)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(33,35,38,44)(34,39,42,43)(36,41,45,46)(37,40,47,48);(2,6)(7,9)(8,12)(15,16)(18,25)(20,24)(26,31)(27,30)(33,36,38,45)(34,40,42,48)(35,41,44,46)(37,39,47,43);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,36)(34,40)(35,41)(37,43)(38,45)(39,47)(42,48)(44,46);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)(33,37,38,47)(34,41,42,46)(35,40,44,48)(36,39,45,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=hi,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, 8653094);