#254210

order	512
nilpotency class	3
isoclinism	isoclinic class of #254179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24223	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=ghi,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, 161406);