#7656827

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

4124 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)(9,10,14,17)(11,18,21,15)(12,13,16,22)(19,24,23,20)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,4,5)(3,6,7,8)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(9,11,14,21)(10,15,17,18)(12,19,16,23)(13,20,22,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,22)(18,24)(21,23);(9,13)(10,16)(11,20)(12,17)(14,22)(15,23)(18,19)(21,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^2=gi,f^2=i,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, 7564023);