#7787739

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

4577 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)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26)(27,34)(28,32)(29,33)(30,35)(31,36)(37,39)(38,40);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,27)(26,31)(28,37)(29,39)(30,36)(32,38)(33,40)(34,35);(10,15)(11,17)(13,21)(20,23)(27,36)(31,34)(37,40)(38,39);(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28)(26,32)(27,37)(29,35)(30,33)(31,38)(34,39)(36,40);(1,3,4,7)(2,6,5,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,30,32)(26,33,35,28)(27,38,36,39)(31,37,34,40);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,34)(28,37)(29,38)(30,36)(31,35)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7694935);