#7736178

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

5007 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,31,29,28);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,26)(24,28)(27,30)(31,32);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(19,26)(20,23)(27,29)(28,31);(1,2)(3,6)(4,5)(7,8)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,32,29);(1,3,4,8)(2,6,5,7)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,32)(29,30);(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643374);