#8701694

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

4241 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(25,26)(27,33)(28,39)(29,34)(30,38)(31,36)(32,37)(35,40);(2,6)(3,7)(10,15)(11,18)(17,24)(19,21)(25,27,32,31)(26,33,37,36)(28,35,29,38)(30,39,40,34);(1,3,5,7)(2,4,6,8)(25,28,32,29)(26,34,37,39)(27,38,31,35)(30,33,40,36);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,29,32,28)(26,34,37,39)(27,38,31,35)(30,36,40,33);(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,30,32,40)(26,35,37,38)(27,39,31,34)(28,33,29,36);(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,31,32,27)(26,36,37,33)(28,35,29,38)(30,39,40,34)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=gi,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, 8608890);