#8125658

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

4766 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8032854);