#8435092

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,e^2=h,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, 8342288);