#9923652

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

4992 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830848);