#8799150

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	#43897	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 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)(11,12)(13,14)(15,16)(17,18,23,27)(19,21,25,32)(20,26,22,24)(28,31,29,30);(1,2)(3,8)(4,5)(6,7)(10,14)(12,15)(17,19)(18,21)(20,28)(22,29)(23,25)(24,30)(26,31)(27,32);(2,7)(3,4)(10,14)(11,16)(17,20,23,22)(18,24,27,26)(19,28,25,29)(21,30,32,31);(1,3,6,4)(2,8,7,5)(9,11,13,16)(10,15,14,12);(1,4,6,3)(2,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,25)(19,27)(20,30)(22,31)(23,32)(24,29)(26,28);(1,5)(2,3)(4,7)(6,8)(9,12,13,15)(10,16,14,11)(17,22,23,20)(18,26,27,24)(19,29,25,28)(21,31,32,30)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706346);