#8115520

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

4064 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,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, 8022716);