#8152190

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
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8059386);