#8424761

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

4546 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,e^2=g,f^2=h,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, 8331957);