#8345871

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

4198 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:

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

Polycyclic presentation:

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

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, 8253067);