#7744214

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651410);