#7766353

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	#59811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4325 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(27,36)(29,39)(32,37)(33,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,26,31,35)(27,37,36,32)(28,30,34,40)(29,33,39,38);(11,20)(13,21)(15,19)(17,24)(25,27,31,36)(26,32,35,37)(28,39,34,29)(30,33,40,38);(9,12)(10,16)(11,21)(13,20)(14,23)(15,17)(18,22)(19,24)(25,28,31,34)(26,30,35,40)(27,29,36,39)(32,38,37,33);(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,29)(26,33)(27,28)(30,37)(31,39)(32,40)(34,36)(35,38);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,18,24)(11,12,20,23)(15,22,19,16)(25,30)(26,34)(27,38)(28,35)(29,37)(31,40)(32,39)(33,36)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7673549);