#8930695

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

4575 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837891);