#7737409

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	#48910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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)(25,26)(27,35)(28,34)(29,32)(30,36)(31,33)(37,38)(39,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,31,29)(26,32,33,35)(28,37,36,39)(30,38,34,40);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,19)(18,23)(26,33)(28,36)(32,35)(37,39);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,36)(26,34,33,30)(27,37,29,39)(32,40,35,38);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29,31,27)(26,35,33,32)(28,39,36,37)(30,40,34,38);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,34)(26,36,33,28)(27,38,29,40)(32,39,35,37)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644605);