#8131229

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

4385 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038425);