#8722192

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

4969 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^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, 8629388);