#7653334

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4621 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:

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

Polycyclic presentation:

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

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, 7560530);