#8696208

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

4641 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603404);