#8701601

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=hi,e^2=g,f^2=gi,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, 8608797);