#8891778

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

4813 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)(3,7)(4,8)(5,6)(17,18,22,23)(19,24,27,31)(20,21,30,28)(25,32,29,26);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,27,31)(20,28,30,21)(25,26,29,32);(2,6)(4,7)(10,13)(14,16)(18,23)(21,28)(24,31)(26,32);(2,6)(3,8)(11,15)(14,16)(17,19,22,27)(18,24,23,31)(20,29,30,25)(21,26,28,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,21)(19,25)(22,30)(23,28)(24,32)(26,31)(27,29);(1,4,5,7)(2,3,6,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,20)(19,26)(22,28)(23,30)(24,29)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,f^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, 8798974);