#7874220

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	#37714	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4795 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)(3,6)(4,7)(5,8)(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,25,27,23)(20,24,29,28)(26,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32);(1,2,5,8)(3,4,7,6)(9,10,13,12)(11,14,16,15)(17,18,22,21)(19,25,27,23)(20,24,29,28)(26,31,32,30);(1,3)(2,6)(4,8)(5,7)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30);(1,4)(2,7)(3,8)(5,6)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=hi,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7781416);