#8088300

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

4727 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)(9,10,13,12)(11,15,16,14)(17,18,23,25)(19,24,20,22)(21,26,30,32)(27,31,28,29);(1,3,6,4)(2,5,8,7)(9,11,13,16)(10,14,12,15)(17,19,23,20)(18,24,25,22)(21,27,30,28)(26,31,32,29);(1,2,6,8)(3,7,4,5)(9,10,13,12)(11,14,16,15)(18,25)(22,24)(26,32)(29,31);(3,4)(5,7)(9,10,13,12)(11,14,16,15)(17,20,23,19)(18,22,25,24)(21,27,30,28)(26,31,32,29);(1,4,6,3)(2,7,8,5)(9,10,13,12)(11,14,16,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32);(1,5,6,7)(2,3,8,4)(9,12,13,10)(11,15,16,14)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=ghi,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, 7995496);