#8115519

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=gh,e^2=h,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,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022715);