#7650787

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

4998 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)(10,13)(11,15)(17,18,22,23)(19,26,29,24)(20,21,25,31)(27,28,30,32);(1,2,5,6)(3,7,4,8)(10,13)(14,16);(9,10)(11,14)(12,13)(15,16)(18,23)(21,31)(24,26)(28,32);(9,11,12,15)(10,14,13,16)(17,19,22,29)(18,24,23,26)(20,30,25,27)(21,28,31,32);(1,3,5,4)(2,7,6,8)(17,20,22,25)(18,21,23,31)(19,27,29,30)(24,32,26,28);(1,4,5,3)(2,8,6,7)(17,21)(18,25)(19,28)(20,23)(22,31)(24,27)(26,30)(29,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7557983);