#8195955

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	#59222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4401 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8103151);