#7987490

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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)(9,10,15,17)(11,19,13,16)(12,14,20,18)(21,22,23,24)(25,26)(27,33)(28,30)(29,39)(31,34)(32,35)(36,40)(37,38);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,24,18,22)(25,27,32,29)(26,33,35,39)(28,38,31,36)(30,37,34,40);(10,17)(11,13)(12,20)(22,24)(25,28,32,31)(26,30,35,34)(27,36,29,38)(33,40,39,37);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,22,19,24)(25,29,32,27)(26,33,35,39)(28,38,31,36)(30,40,34,37);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,22,18,24)(25,30)(26,28)(27,37)(29,40)(31,35)(32,34)(33,38)(36,39);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,21,19,23)(25,31,32,28)(26,34,35,30)(27,38,29,36)(33,37,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=ghi,e^2=g,f^2=i,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, 7894686);