#8679900

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	#36231	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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)(6,8)(11,19)(15,17)(20,24)(21,23)(25,26,31,34)(27,29,37,32)(28,38,30,33)(35,40,36,39);(3,7)(6,8)(9,10)(11,17)(12,13)(14,18)(15,19)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,34,29)(28,35,30,36)(33,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,18)(22,24)(25,28,31,30)(26,33,34,38)(27,35,37,36)(29,39,32,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,22)(10,16,18,13)(11,20,19,24)(15,23,17,21)(25,26)(27,29)(28,33)(30,38)(31,34)(32,37)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,20)(13,23)(14,19)(15,18)(16,21)(22,24)(25,29)(26,27)(28,39)(30,40)(31,32)(33,35)(34,37)(36,38);(9,13,14,16)(10,12,18,22)(11,21,19,23)(15,20,17,24)(25,30,31,28)(26,33,34,38)(27,36,37,35)(29,39,32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,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, 8587096);