#7657754

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

4581 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)(9,10)(11,17)(12,16)(13,18)(14,19)(15,22)(20,21)(23,24)(28,36)(31,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,12,14,18)(10,16,19,13)(11,20,22,24)(15,23,17,21)(25,28,29,36)(26,31,33,38)(27,34,35,40)(30,37,32,39);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,20)(13,23)(14,22)(15,19)(16,21)(18,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(9,13,14,16)(10,18,19,12)(11,21,22,23)(15,24,17,20)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

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

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, 7564950);