#8438542

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

4957 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345738);