#7652374

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4660 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559570);