#7787290

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

4995 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7694486);