#7652388

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5296 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,29)(21,26)(22,27)(23,25)(28,31)(30,32);(2,6)(3,8)(9,10,12,14)(11,13,16,15)(17,19,23,20)(18,24,25,29)(21,27,31,30)(22,28,32,26);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(18,25)(22,32)(24,29)(26,28);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,19)(18,24,25,29)(21,27,31,30)(22,26,32,28);(17,21,23,31)(18,26,25,28)(19,27,20,30)(22,29,32,24);(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,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, 7559584);