#7645309

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

6212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7552505);