#8127008

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

4951 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=gh,f^2=g,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^c=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, 8034204);