#7715170

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622366);