#7981540

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7888736);