#8695297

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=gh,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, 8602493);