#7653784

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	#34964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4777 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 7560980);