#8081085

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

4862 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,29)(30,32);(9,11,13,16)(10,14,12,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,32,30,31);(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,16,15)(18,20)(21,29)(23,25)(26,30);(3,7)(5,8)(9,10,13,12)(11,14,16,15)(17,19,22,27)(18,23,20,25)(21,30,29,26)(24,32,28,31);(1,3)(2,5)(4,7)(6,8)(9,10,13,12)(11,14,16,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,32,30,31);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,21,22,29)(18,24,20,28)(19,26,27,30)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^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, 7988281);