#8903122

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

5335 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,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, 8810318);