#8267616

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	#45011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8174812);