#253945

order	512
nilpotency class	3
isoclinism	isoclinic class of #253891
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	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 860 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,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, 161141);