#8702594

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

4434 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609790);