#8398564

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

5345 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=g,f^2=gi,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, 8305760);