#10320440

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

5279 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227636);