#478420

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
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	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 983 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^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, 385616);