#8406690

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

5888 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:

(1,2,6,7)(3,5,4,8)(10,13)(14,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31);(2,7)(3,4)(11,15)(14,16)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(19,28)(22,31)(24,32)(26,29);(1,4,6,3)(2,5,7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31);(1,5)(2,3)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(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|a^2=g,b^2=gi,c^2=g,d^2=h,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.

Relationships

maximal subgroup of	#72336231
maximal normal subgroup of	#72336231
maximal quotient of	#72336356

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313886);