#7732349

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	#35569	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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)(9,10,15,14)(11,18,13,16)(12,17,22,21)(19,24,20,23)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,14,18)(12,19,22,20)(17,23,21,24)(27,30)(32,34)(35,37)(39,40);(11,13)(16,18)(19,20)(23,24)(25,27,31,30)(26,32,28,34)(29,37,38,35)(33,40,36,39);(1,3,4,7)(2,6,5,8)(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,13,15,11)(10,16,14,18)(12,20,22,19)(17,23,21,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,14,15,10)(11,16,13,18)(12,21,22,17)(19,23,20,24)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7639545);