#7650260

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	#39408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4627 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=i,f^2=h,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, 7557456);