#8724876

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

5071 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=gi,e^2=g,f^2=gi,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, 8632072);