#10196166

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

4129 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103362);