#7905683

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

5121 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7812879);