#8429638

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

5080 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)(3,6)(4,7)(5,8)(10,13)(14,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32);(3,8)(5,6)(11,15)(14,16)(18,24)(19,20)(26,29)(27,28);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30);(9,10)(11,14)(12,13)(15,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32);(1,2)(3,5)(4,7)(6,8)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,f^2=g,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, 8336834);