#7734382

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×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7641578);