#8115845

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

4888 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8023041);