#7651931

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	#38909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4471 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,20,21)(12,23,18,14)(13,24,17,19);(11,22)(14,23)(16,21)(19,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(9,12)(10,17)(11,23)(13,20)(14,22)(15,18)(16,19)(21,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(9,13,15,17)(10,18,20,12)(11,24,22,19)(14,21,23,16)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,3,4,7)(2,6,5,8)(9,14,15,23)(10,19,20,24)(11,12,22,18)(13,21,17,16)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=hi,f^2=gh,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, 7559127);