#7891237

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

4828 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)(3,8)(4,6)(5,7)(10,14)(15,16)(17,18)(19,24)(20,22)(21,31)(23,27)(25,30)(26,28)(29,32);(1,3,5,6)(2,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,23,29)(18,24,27,32)(20,28,30,21)(22,26,25,31);(1,2)(3,8)(4,6)(5,7)(17,20,23,30)(18,25,27,22)(19,21,29,28)(24,26,32,31);(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,28)(21,29)(23,30)(24,26)(25,27)(31,32);(1,2)(3,4)(5,7)(6,8)(9,11,13,12)(10,15,14,16)(17,21,23,28)(18,26,27,31)(19,20,29,30)(22,24,25,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,22)(18,20)(19,26)(21,32)(23,25)(24,28)(27,30)(29,31)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=ghi,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, 7798433);