#7741513

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	#51333	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7648709);