#8522140

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

5055 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8429336);