#253934

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	#24227	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 850 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)(3,4)(5,7)(6,8)(9,10,13,15)(11,16,21,28)(12,22,23,17)(14,26,25,18)(19,29,27,31)(20,30,34,35)(24,32,37,36)(33,39,38,40);(1,3,6,7)(2,4,8,5)(10,15)(11,19,21,27)(16,29,28,31)(17,22)(18,26)(20,33,34,38)(30,39,35,40)(32,36);(2,3)(4,5)(7,8)(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);(9,12,13,23)(10,17,15,22)(11,20,21,34)(14,24,25,37)(16,30,28,35)(18,32,26,36)(19,33,27,38)(29,39,31,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=h,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, 161130);