#9838194

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	#44342	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4626 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=ghi,f^2=h,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, 9745390);