#8614125

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=i,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, 8521321);