#7655853

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	#35166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5086 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7563049);