#8522139

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

5074 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,12,15,16)(18,23)(19,25)(24,30)(26,31);(2,3)(6,7)(10,14)(11,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,15)(14,16)(20,27)(24,30)(26,31)(28,32);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32);(1,4)(2,6)(3,7)(5,8)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32);(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8429335);