#7983154

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

4643 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=g,f^2=ghi,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, 7890350);