#7745343

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4744 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)(10,13)(14,16)(17,18,23,25)(19,21,20,24)(22,26,30,32)(27,29,28,31);(9,10,12,13)(11,14,15,16)(17,19,23,20)(18,24,25,21)(22,27,30,28)(26,31,32,29);(2,6)(3,7)(18,25)(19,20)(22,30)(29,31);(1,3,5,7)(2,4,6,8)(17,20,23,19)(18,24,25,21)(22,28,30,27)(26,31,32,29);(1,4)(2,7)(3,6)(5,8)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,19,25,20)(22,29,30,31)(26,27,32,28);(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,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, 7652539);