#8459231

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	#34715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4771 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8366427);