#8361774

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

4841 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8268970);