#8267596

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	#59053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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)(9,10,15,19)(11,20,14,16)(12,18,13,17)(21,23,22,24);(3,7)(6,8)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(25,26)(27,30)(28,32)(29,37)(31,34)(33,39)(35,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,29,31,39)(26,33,34,37)(27,35,28,36)(30,38,32,40);(9,14,15,11)(10,16,19,20)(12,21,13,22)(17,24,18,23)(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=gi,d^2=hi,e^2=i,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, 8174792);