#7656357

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	#42794	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563553);