#8657127

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

5010 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)(3,8)(4,6)(5,7)(9,10,12,13)(11,15,16,14)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31);(2,4)(7,8)(10,13)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(21,30)(24,32)(26,29)(28,31);(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29);(1,5,6,3)(2,7,4,8)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8564323);