#8109987

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

4810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017183);