#9446287

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

4976 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)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32);(1,2)(3,6)(4,5)(7,8)(10,13)(14,15)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32);(1,3,4,7)(2,6,5,8)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32);(1,2)(3,6)(4,5)(7,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9353483);