#8678983

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

4346 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)(3,7)(9,10)(11,18)(12,17)(13,14)(15,19)(16,23)(20,22)(21,24)(25,26,31,34)(27,29,37,32)(28,38,30,33)(35,40,36,39);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,31,37)(26,32,34,29)(28,35,30,36)(33,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,24,18,22)(25,28,31,30)(26,33,34,38)(27,35,37,36)(29,39,32,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,24,18,22)(25,26)(27,29)(28,33)(30,38)(31,34)(32,37)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,18)(12,20)(13,21)(14,24)(15,23)(16,19)(17,22)(25,29)(26,27)(28,39)(30,40)(31,32)(33,35)(34,37)(36,38);(9,14,15,17)(10,13,19,12)(11,22,23,24)(16,21,18,20)(25,30,31,28)(26,33,34,38)(27,36,37,35)(29,39,32,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=gh,f^2=gi,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, 8586179);