#8899055

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

5590 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)(3,4)(5,6)(7,8)(17,18)(19,24)(20,30)(21,25)(22,28)(23,27)(26,32)(29,31);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,31)(20,28,21,26)(22,25,32,30);(2,6)(4,8)(10,13)(14,16)(17,20,23,21)(18,25,27,30)(19,26,29,28)(22,31,32,24);(2,6)(3,7)(11,15)(14,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,32)(29,31);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,25,27,30)(19,28,29,26)(22,31,32,24);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,26)(19,25)(20,24)(21,31)(23,32)(27,28)(29,30)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=gi,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, 8806251);