#7643485

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4832 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)(10,14)(15,16)(18,24)(22,26)(25,27)(29,31);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,24)(19,27,20,25)(21,26,32,22)(28,31,30,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,19,23,20)(18,25,24,27)(21,30,32,28)(22,29,26,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,23,19)(18,25,24,27)(21,30,32,28)(22,31,26,29);(17,21)(18,26)(19,28)(20,30)(22,24)(23,32)(25,29)(27,31);(17,22,23,26)(18,21,24,32)(19,29,20,31)(25,30,27,28)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,f^2=hi,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, 7550681);