#8696277

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	#59099	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603473);