#8913956

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

4920 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821152);