#8505278

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

3978 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412474);