#8159884

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

4822 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)(3,5)(4,7)(6,8)(10,13)(11,12)(17,18,23,26)(19,27,30,24)(20,31,29,21)(22,28,32,25);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,16,12,15)(17,19,23,30)(18,24,26,27)(20,22,29,32)(21,28,31,25);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,15,13,16)(17,18,23,26)(19,24,30,27)(20,31,29,21)(22,25,32,28);(2,3)(6,7)(9,12,14,11)(10,16,13,15)(17,20,23,29)(18,21,26,31)(19,22,30,32)(24,28,27,25);(1,3,5,2)(4,7,8,6)(9,13,14,10)(11,16,12,15)(17,21)(18,20)(19,28)(22,24)(23,31)(25,30)(26,29)(27,32);(1,4)(2,6)(3,7)(5,8)(17,22)(18,25)(19,29)(20,30)(21,24)(23,32)(26,28)(27,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8067080);