#130248

order	512
nilpotency class	4
isoclinism	isoclinic class of #130246
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,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, 37444);