#7660942

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	#52163	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4798 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7568138);