#8702297

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4601 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8609493);