#8662838

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	#53174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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)(9,10,15,18)(11,17,22,16)(12,14,13,23)(19,24,20,21)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,29,34)(26,30,32,37)(28,33,35,39)(31,36,38,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,13)(10,14,18,23)(11,19,22,20)(16,21,17,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(2,4)(6,7)(9,13,15,12)(10,14,18,23)(11,20,22,19)(16,21,17,24)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,17)(12,19)(13,20)(14,24)(15,22)(16,18)(21,23)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,4,5,2)(3,7,8,6)(9,14)(10,13)(11,21)(12,18)(15,23)(16,20)(17,19)(22,24)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,36,37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,f^2=h,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, 8570034);