#7642127

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

5786 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549323);