#8799479

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

4832 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)(17,18,23,25)(19,29,31,24)(20,32,30,26)(21,27,28,22);(1,2)(3,8)(4,5)(6,7)(9,10,13,12)(11,15,16,14)(17,19,23,31)(18,24,25,29)(20,28,30,21)(22,32,27,26);(2,7)(3,4)(9,11,13,16)(10,14,12,15)(18,25)(20,30)(21,28)(24,29);(1,3,6,4)(2,8,7,5)(9,12,13,10)(11,15,16,14)(17,20,23,30)(18,26,25,32)(19,28,31,21)(22,29,27,24);(1,4,6,3)(2,5,7,8)(17,21,23,28)(18,27,25,22)(19,30,31,20)(24,32,29,26);(1,5)(2,3)(4,7)(6,8)(17,22,23,27)(18,28,25,21)(19,26,31,32)(20,29,30,24)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=hi,e^2=gi,f^2=h,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, 8706675);