#7826519

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5505 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16);(1,3)(2,6)(4,7)(5,8)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31);(2,7)(4,6)(9,10,12,14)(11,13,16,15)(18,23)(21,30)(24,31)(27,32);(1,4,5,6)(2,3,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,26)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,25)(18,21,23,30)(19,26,28,29)(24,32,31,27);(17,21)(18,25)(19,27)(20,23)(22,30)(24,26)(28,32)(29,31)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7733715);