#8340079

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

4701 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)(17,18,23,22)(19,26,25,21)(20,24,28,30)(27,32,31,29);(1,2,6,4)(3,8,5,7)(10,13)(14,15)(20,28)(24,30)(27,31)(29,32);(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,31)(30,32);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32);(1,4,6,2)(3,7,5,8)(9,11,12,16)(10,14,13,15)(17,21,23,26)(18,25,22,19)(20,29,28,32)(24,31,30,27);(1,5,6,3)(2,7,4,8)(17,22,23,18)(19,26,25,21)(20,30,28,24)(27,32,31,29)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8247275);