#8751497

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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)(3,8,4,6)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,3,5,4)(2,6,7,8)(10,14)(11,18)(16,20)(19,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(3,4)(6,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=ghi,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, 8658693);