#8131175

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

4496 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)(9,10,14,18)(11,12,16,15)(13,21,22,17)(19,23,24,20)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(3,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(26,32)(27,28)(34,37)(35,36);(1,2,5,4)(3,6,7,8)(11,16)(12,15)(19,24)(20,23)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3,5,7)(2,6,4,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,4,5,2)(3,8,7,6)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ghi,e^2=ghi,f^2=g,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, 8038371);