#8202252

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,3)(2,6)(4,7)(5,8)(11,19)(12,22)(15,24)(16,21)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,2,5,4)(3,7,8,6)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,2)(3,6,8,7)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37);(1,3)(2,7)(4,6)(5,8)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)

Polycyclic presentation:

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

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, 8109448);