#7719075

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

4077 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)(10,16)(11,13)(17,23)(19,20)(29,37)(32,39)(34,40)(36,38);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,23)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(9,11,15,13)(10,14,16,18)(12,20,22,19)(17,24,23,21)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(9,12,15,22)(10,17,16,23)(11,19,13,20)(14,21,18,24);(9,13,15,11)(10,18,16,14)(12,20,22,19)(17,24,23,21)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,11,16,13)(12,21,22,24)(17,19,23,20)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

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, 7626271);