#7782731

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,23,21,26)(20,25,30,24)(28,32,29,31);(9,11,12,16)(10,13,14,15)(17,19,22,21)(18,23,27,26)(20,29,30,28)(24,32,25,31);(2,5)(6,7)(17,20,22,30)(18,24,27,25)(19,28,21,29)(23,31,26,32);(1,2)(3,6)(4,5)(7,8)(17,20)(18,25)(19,29)(21,28)(22,30)(23,31)(24,27)(26,32);(1,3)(2,6)(4,8)(5,7)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32);(17,21,22,19)(18,26,27,23)(20,28,30,29)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

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, 7689927);