#8424896

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

4416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332092);