#8120011

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8027207);