#8505279

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

4900 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)(3,4)(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,18,23,22)(19,27,29,24)(20,25,31,30)(21,26,28,32);(2,6)(3,8)(10,11)(14,15)(17,18,23,22)(19,27,29,24)(20,30,31,25)(21,32,28,26);(9,11,13,10)(12,15,16,14)(17,19,23,29)(18,24,22,27)(20,28,31,21)(25,26,30,32);(2,6)(4,7)(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,28)(21,29)(22,30)(23,31)(24,26)(27,32);(1,3)(2,7)(4,6)(5,8)(17,21,23,28)(18,26,22,32)(19,20,29,31)(24,30,27,25);(1,4,5,7)(2,3,6,8)(17,22,23,18)(19,27,29,24)(20,25,31,30)(21,32,28,26)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412475);