#8081025

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

4866 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7988221);