#8438114

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

5012 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gh,e^2=gi,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, 8345310);