#130699

order	512
nilpotency class	4
isoclinism	isoclinic class of #130482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#555 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37895);