#7883702

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

4825 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790898);