#10412363

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

5166 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319559);