#8197633

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
prime radical	#36096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4953 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^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

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, 8104829);