#7645979

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

4416 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7553175);