#7658810

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	#37440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4864 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)(9,10,12,14)(11,15,16,13)(18,24)(20,29)(25,26)(27,31);(1,2,4,5)(3,6,7,8)(11,16)(13,15)(17,18,23,24)(19,26,21,25)(20,30,29,22)(27,28,31,32);(9,11)(10,13)(12,16)(14,15)(17,19,23,21)(18,25,24,26)(20,27,29,31)(22,28,30,32);(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,27)(21,31)(23,29)(24,30)(25,28)(26,32);(1,3,4,7)(2,6,5,8)(17,21,23,19)(18,25,24,26)(20,31,29,27)(22,28,30,32);(17,22)(18,20)(19,28)(21,32)(23,30)(24,29)(25,27)(26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,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, 7566006);