#7657701

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,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, 7564897);