#8110619

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017815);