#7981534

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4531 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=h,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, 7888730);