#7709500

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	C₂×#540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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)(3,5,7,8)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26,31,28)(27,34,33,30)(29,36,37,32)(35,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(10,12)(16,17)(18,20)(23,24)(26,28)(27,33)(29,37)(38,40);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,26)(27,34,33,30)(29,36,37,32)(35,40,39,38);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,37)(26,32,28,36)(27,35,33,39)(30,38,34,40);(1,3,4,7)(2,5,6,8)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30,31,34)(26,33,28,27)(29,38,37,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616696);