#8389319

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

5038 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)(14,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,31)(29,32);(1,2,5,6)(3,7,8,4)(11,15)(14,16)(17,18,22,25)(19,23,20,27)(21,24,29,32)(26,30,28,31);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,25,27)(21,28,29,26)(24,31,32,30);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,23,25,27)(21,28,29,26)(24,30,32,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,31)(29,32);(1,4)(2,3)(5,7)(6,8)(17,21,22,29)(18,24,25,32)(19,26,20,28)(23,30,27,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8296515);