#479878

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

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

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

normal subgroups only

This group has 535 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,4)(3,6,7,8)(9,10,15,21,16,22,14,20)(11,23,25,17,26,28,18,12)(13,19,30,35,32,37,31,36)(24,38,39,27,40,29,34,33);(2,4)(6,8)(9,11,14,25,16,26,15,18)(10,17,20,28,22,12,21,23)(13,24,31,39,32,40,30,34)(19,33,36,38,37,27,35,29);(1,3,5,7)(2,6,4,8)(9,12)(10,18)(11,20)(13,29)(14,23)(15,28)(16,17)(19,24)(21,26)(22,25)(27,30)(31,33)(32,38)(34,35)(36,39)(37,40);(1,4,5,2)(3,6,7,8)(9,13)(10,19)(11,24)(12,27)(14,31)(15,30)(16,32)(17,33)(18,34)(20,36)(21,35)(22,37)(23,29)(25,39)(26,40)(28,38)

Polycyclic presentation:

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

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, 387074);