#7766079

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

4437 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)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(27,35)(29,37)(32,36)(34,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,26,31,30)(27,36,35,32)(28,33,39,38)(29,34,37,40);(11,18)(15,22)(19,23)(20,24)(25,27,31,35)(26,32,30,36)(28,37,39,29)(33,34,38,40);(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28,31,39)(26,33,30,38)(27,29,35,37)(32,40,36,34);(1,3,5,4)(2,7,6,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,36)(35,39);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30,31,26)(27,32,35,36)(28,38,39,33)(29,40,37,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=gh,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, 7673275);