#8807323

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

5086 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714519);