#8479487

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

4891 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)(4,7)(5,6)(10,13)(11,12)(20,28)(23,29)(27,31)(30,32);(2,6)(3,7)(9,10,14,13)(11,16,12,15)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,30)(31,32);(9,11,14,12)(10,15,13,16)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,30,29,32);(2,6)(4,8)(9,12,14,11)(10,16,13,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,29)(31,32);(1,3,5,7)(2,4,6,8)(9,13,14,10)(11,16,12,15)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,29)(26,32);(1,4)(2,7)(3,6)(5,8)(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,29)(28,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=g,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, 8386683);