#8718520

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	#37833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625716);