#8425204

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	#53014	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332400);