#8662871

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	#36530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570067);