#7658638

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

4310 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565834);