#8088158

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

4733 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,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, 7995354);