#9937888

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

4960 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=g,e^2=g,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, 9845084);