#8977258

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 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)(3,7)(4,8)(5,6)(25,26)(27,32)(28,34)(29,30)(31,36)(33,37)(35,40)(38,39);(2,6)(3,8)(9,10,14,15)(11,16,19,24)(12,17,22,21)(13,18,23,20)(25,27,31,38)(26,32,36,39)(28,35,37,29)(30,34,40,33);(1,3)(2,4)(5,8)(6,7)(10,15)(11,19)(12,22)(13,23)(25,28,31,37)(26,33,36,34)(27,29,38,35)(30,32,40,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,18)(13,21)(14,19)(15,24)(17,23)(20,22)(25,28)(26,34)(27,35)(29,38)(30,39)(31,37)(32,40)(33,36);(1,2)(3,4)(5,6)(7,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,19)(23,24)(25,29,31,35)(26,30,36,40)(27,37,38,28)(32,33,39,34);(1,4)(2,3)(5,7)(6,8)(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,19)(22,24)(25,30)(26,35)(27,34)(28,39)(29,36)(31,40)(32,37)(33,38)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8884454);