#8622117

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

4620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8529313);