#7737815

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,3,4,7)(2,5,6,8)(10,16)(12,18)(17,21)(20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22);(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7645011);