#8933844

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

4948 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841040);