#8679877

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587073);