#9848962

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

4586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=h,e^2=ghi,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, 9756158);