#7658939

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	#37606	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4602 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)(6,8)(9,10)(11,15)(12,14)(13,16)(20,28)(23,29)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,30)(31,32);(9,12,14,10)(11,15,13,16)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,30,29,32);(9,13,14,11)(10,15,12,16)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,29)(26,32);(1,3,4,7)(2,6,5,8)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,29)(31,32);(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,29)(28,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566135);