#345822

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 612 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)(11,20,25,22)(12,26,30,35)(13,31,14,18)(16,24,33,23)(17,28,36,29)(21,32,39,40)(27,34,38,37);(1,2,4,7)(3,5,8,6)(9,11,13,23,15,25,14,24)(10,16,18,20,19,33,31,22)(12,27,29,21,30,38,28,39)(17,34,26,40,36,37,35,32);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,28)(14,29)(15,30)(16,32)(18,35)(19,36)(20,37)(22,34)(23,27)(24,38)(25,39)(26,31)(33,40);(9,10)(11,22)(12,17)(13,31)(14,18)(15,19)(16,24)(20,25)(21,34)(23,33)(26,28)(27,40)(29,35)(30,36)(32,38)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253018);