#8702924

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4486 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8610120);