#130013

order	512
nilpotency class	4
isoclinism	isoclinic class of #129769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 578 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,f^c=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, 37209);