#394054

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14715
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3877 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)(3,15)(4,17)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,27)(16,28)(19,22)(20,30)(21,26)(24,31)(29,32);(1,3)(2,9)(4,21)(5,17)(6,19)(7,18)(8,20)(10,26)(11,24)(12,30)(13,15)(14,22)(16,23)(25,29)(27,32)(28,31);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,28)(11,19)(12,17)(13,31)(14,32)(15,27)(18,21)(20,25)(26,30);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,24)(10,19)(12,16)(13,25)(14,21)(15,29)(18,32)(20,31)(23,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215968, #72219683, #72219688, …
maximal normal subgroup of	#72215968, #72219683, #72219688, …
maximal quotient of	C₂×#394054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301250);