#8934580

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

4952 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gh,e^2=gh,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, 8841776);