#8898093

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

4880 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)(3,5,7,8)(17,18,22,23)(19,27,29,24)(20,25,30,32)(21,28,31,26);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(18,23)(20,30)(24,27)(26,28);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(18,23)(21,31)(24,27)(25,32);(11,15)(13,16)(17,19,22,29)(18,24,23,27)(20,26,30,28)(21,25,31,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,30)(18,25,23,32)(19,28,29,26)(21,27,31,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,26)(19,25)(20,27)(22,31)(23,28)(24,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805289);