#7631255

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538451);