#7745352

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

4810 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

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, 7652548);