#8351680

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,e^2=gh,f^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,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, 8258876);