#268708

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175904);