#9836138

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

5004 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^2=gi,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, 9743334);