#9445249

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9352445);