#7979103

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

5244 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7886299);