#8700479

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

5775 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312150, #72312209, #72312307, …
maximal normal subgroup of	#72312150, #72312209, #72312307, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607675);