#469243

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,9,12,11,16,10,15)(7,17,19,13,21,27,20,29)(8,14,24,26,22,32,25,30)(18,33,23,36,34,31,35,28);(6,12,16,15)(8,22)(13,27,29,17)(14,30,32,26)(18,34)(23,35)(24,25)(28,31,36,33);(1,2,4,3)(5,7)(6,13)(8,23)(9,20)(10,19)(11,21)(12,17)(14,31)(15,27)(16,29)(18,24)(22,35)(25,34)(26,36)(28,30)(32,33);(1,3,4,2)(5,8,10,25,11,22,9,24)(6,14,15,30,16,32,12,26)(7,18,20,35,21,34,19,23)(13,28,17,31,29,36,27,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376439);