#8936771

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

5033 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gh,f^2=h,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, 8843967);