#8689237

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
prime radical	#41962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4493 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^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, 8596433);