#7645329

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

5123 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)(10,13)(11,15)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31);(1,2)(3,6)(4,7)(5,8)(10,13)(14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30);(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(1,3,5,4)(2,6,8,7)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30);(1,4,5,3)(2,7,8,6)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,e^2=g,f^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, 7552525);