#7736109

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4268 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643305);