#7665657

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4731 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,13,16,15)(18,22)(19,25)(20,28)(30,32);(1,3,6,4)(2,7,5,8)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,2,6,5)(3,7,4,8)(9,11)(10,15)(12,13)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(9,10,14,12)(11,15,16,13)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,16,13);(1,5,6,2)(3,8,4,7)(9,13,14,15)(10,11,12,16)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572853);