#7710516

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

5242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7617712);