#479649

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

579 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fg,c^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,d^c=de,f^c=fi,f^d=fi

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, 386845);