#8182056

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

5830 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)(3,5)(4,6)(7,8)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,31)(29,30);(2,3)(6,7)(9,10,13,14)(11,15,16,12)(17,19,23,20)(18,24,25,26)(21,28,32,30)(22,29,27,31);(4,8)(6,7)(10,14)(12,15)(18,25)(22,27)(24,26)(29,31);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20,23,19)(18,26,25,24)(21,30,32,28)(22,31,27,29);(17,21)(18,27)(19,28)(20,30)(22,25)(23,32)(24,31)(26,29);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,22,23,27)(18,21,25,32)(19,29,20,31)(24,28,26,30)

Polycyclic presentation:

2^9|b^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8089252);