#8828708

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

4825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735904);