#507733

order	512
nilpotency class	4
isoclinism	isoclinic class of #499946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi

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, 414929);