#8749685

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	#59392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4238 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)(11,13)(12,16)(19,21)(20,22)(26,30)(28,35)(31,36)(34,39);(1,2)(3,4)(5,7)(6,8)(9,10,15,18)(11,12,13,16)(14,17,23,24)(19,20,21,22)(27,33)(31,36)(34,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,22,24,20)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,8)(2,6,7,4)(9,12,15,16)(10,11,18,13)(14,22,23,20)(17,21,24,19)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,2)(3,6)(4,8)(5,7)(9,13,15,11)(10,16,18,12)(14,19,23,21)(17,20,24,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,6)(2,3,7,8)(9,14,15,23)(10,17,18,24)(11,19,13,21)(12,20,16,22)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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, 8656881);