#8613228

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

5558 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8520424);