#8749801

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8656997);