#8438116

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

4949 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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, 8345312);