#345864

order	512
nilpotency class	3
isoclinism	isoclinic class of #345848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

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, 253060);