#7642487

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
prime radical	#37860	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5018 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549683);