#8463494

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

4781 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=g,e^2=i,f^2=hi,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, 8370690);