#7649925

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

4508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7557121);