#8058715

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

4954 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)(10,14)(15,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,30)(27,31)(28,32);(3,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28);(1,3)(2,6)(4,7)(5,8)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,16,14,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965911);