#8406692

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

5478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313888);