#8935960

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

5105 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8843156);