#9712824

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
prime radical	#39356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4592 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620020);