#10442545

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

4635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,e^2=gi,f^2=h,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, 10349741);