#8387824

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

5436 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295020);