#7766347

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,e^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, 7673543);