#8463453

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

4415 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)(3,4)(5,7)(6,8)(9,10,15,19)(11,14,18,23)(12,22,21,16)(13,17,20,24)(26,31)(28,37)(32,38)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,18)(10,14,19,23)(12,13,21,20)(16,24,22,17)(27,34)(32,38)(35,40)(36,39);(9,12,15,21)(10,16,19,22)(11,20,18,13)(14,17,23,24)(25,26,30,31)(27,32,34,38)(28,33,37,29)(35,39,40,36);(2,7)(4,6)(9,11)(10,14)(12,20)(13,21)(15,18)(16,17)(19,23)(22,24)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,20)(10,17,19,24)(11,21,18,12)(14,16,23,22)(25,28)(26,33)(27,35)(29,31)(30,37)(32,39)(34,40)(36,38);(1,4,5,6)(2,3,7,8)(9,14)(10,18)(11,19)(12,17)(13,16)(15,23)(20,22)(21,24)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=i,f^2=ghi,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, 8370649);