#8075457

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982653);