#8110534

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

5012 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,15)(14,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30);(2,6)(4,7)(10,11)(14,15)(20,28)(24,31)(27,32)(29,30);(1,3)(2,4)(5,8)(6,7)(9,10,13,11)(12,14,16,15)(17,19,22,25)(18,23,26,21)(20,27,28,32)(24,30,31,29);(1,4)(2,3)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30);(1,4)(2,3)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,3)(2,4)(5,8)(6,7)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,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, 8017730);