#478649

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

793 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 385845);