#7657019

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	#42440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4579 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)(3,6)(4,7)(5,8)(10,14)(15,16)(18,20)(19,27)(25,26)(29,30);(9,10)(11,15)(12,16)(13,14)(17,18,23,20)(19,28,27,24)(21,25,22,26)(29,32,30,31);(1,3)(2,5)(4,8)(6,7)(17,19,23,27)(18,24,20,28)(21,29,22,30)(25,31,26,32);(9,11,13,12)(10,15,14,16)(17,20,23,18)(19,28,27,24)(21,26,22,25)(29,32,30,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,22)(18,25,20,26)(19,29,27,30)(24,31,28,32);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,16,14,15)(17,22,23,21)(18,26,20,25)(19,30,27,29)(24,32,28,31)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=g,f^2=g,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, 7564215);