#254204

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	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 819 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)(11,20)(12,17)(13,16)(14,18)(15,19)(21,30)(22,38)(23,32)(24,37)(25,33)(26,34)(27,31)(28,35)(29,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,16,18,20)(12,23)(15,22,28,27)(17,32)(19,31,35,38)(21,29)(24,25)(26,39)(30,36)(33,37)(34,40);(2,7)(3,8)(4,5)(9,12)(10,17)(11,21)(13,24)(14,26)(15,29)(16,30)(18,34)(19,36)(20,37)(22,39)(23,27)(25,28)(31,40)(32,38)(33,35);(9,13,14,11)(10,16,18,20)(12,24,26,21)(15,27,28,22)(17,30,34,37)(19,31,35,38)(23,25,39,29)(32,36,40,33)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 161400);