#7968067

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875263);