#9836054

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

4932 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9743250);