#8679888

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=h,e^2=gh,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, 8587084);