#419010

order	512
nilpotency class	3
isoclinism	isoclinic class of #419010
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15267
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237065, #72237072, #72237076, …
maximal normal subgroup of	#72237065, #72237072, #72237076, …
maximal quotient of	C₂×#419010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326206);