#7642491

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×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549687);