#8747947

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655143);