#8905546

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

4945 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812742);