#7739263

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	#37269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4912 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7646459);