#8425645

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

4432 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=h,f^2=g,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, 8332841);