#8739363

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=h,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, 8646559);