#8159950

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

5007 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8067146);