#7631281

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	#35261	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5234 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)(3,8,4,7)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,12,15)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29);(1,4,6,3)(2,8,5,7)(9,10,13,12)(11,14,16,15)(18,25)(19,20)(26,30)(27,28);(1,2,6,5)(3,7,4,8)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,15,16,14)(17,21)(18,20)(19,25)(22,29)(23,24)(26,28)(27,30)(31,32);(17,22,23,31)(18,26,25,30)(19,27,20,28)(21,29,24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 7538477);