#8930672

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

4809 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837868);