#8463221

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4766 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,16,12,15)(18,24)(19,20)(25,31)(26,28);(1,3)(2,5)(4,8)(6,7)(11,12)(15,16)(17,18,23,24)(19,22,20,27)(21,25,30,31)(26,29,28,32);(9,11,14,12)(10,15,13,16)(17,19,23,20)(18,22,24,27)(21,26,30,28)(25,29,31,32);(2,6)(5,7)(9,12,14,11)(10,15,13,16)(17,20,23,19)(18,22,24,27)(21,28,30,26)(25,29,31,32);(1,3)(2,5)(4,8)(6,7)(17,21,23,30)(18,25,24,31)(19,26,20,28)(22,29,27,32);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,16,12,15)(17,22)(18,20)(19,24)(21,29)(23,27)(25,28)(26,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=i,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, 8370417);