#7653082

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	#38208	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4870 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=g,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, 7560278);