#8943953

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8851149);