#7898482

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805678);