#7641952

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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)(10,14)(12,19)(15,20)(18,23)(25,26,30,31)(27,32,34,28)(29,37,38,33)(35,36,40,39);(1,2)(3,6)(4,7)(5,8)(11,17)(15,20)(18,23)(22,24)(26,31)(27,34)(33,37)(35,40);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,37)(38,40);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,30,32)(26,27,31,34)(29,36,38,39)(33,35,37,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,37)(38,40);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi

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, 7549148);