#8437784

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
prime radical	#58273	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8344980);