#7641972

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi

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, 7549168);