#7766047

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

4486 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,f^2=h,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, 7673243);