#8924905

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

5030 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832101);