#8699204

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	#45041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8606400);