#8131501

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

4458 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,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, 8038697);