#9938127

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

4781 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,12,16,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,30)(28,32);(1,2,4,6)(3,5,8,7)(9,11,13,16)(10,14,15,12)(18,23)(19,27)(25,32)(26,29);(9,11)(10,12)(13,16)(14,15)(18,23)(19,27)(25,32)(26,29);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,22,27)(18,24,23,30)(20,26,28,29)(21,25,31,32);(1,2)(3,7)(4,6)(5,8)(17,20)(18,25)(19,26)(21,30)(22,28)(23,32)(24,31)(27,29);(1,3)(2,5)(4,8)(6,7)(17,21,22,31)(18,26,23,29)(19,25,27,32)(20,24,28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845323);