#8930534

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

4185 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=gh,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, 8837730);