#8698923

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,f^2=gi,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, 8606119);