#9105263

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9012459);