#124815

order	512
nilpotency class	4
isoclinism	isoclinic class of #124708
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,10,13,17,16,19,24,33,15,21,25,34,14,20,11,22)(12,26,28,39,30,38,23,32,27,36,31,40,29,18,35,37);(1,2,4,6)(3,8,7,5)(9,11,15,24)(10,17)(12,23,27,35)(13,16,25,14)(18,32)(19,22)(20,33)(21,34)(26,39)(28,29,31,30)(36,40)(37,38);(9,12)(10,18,21,38)(11,23)(13,31)(14,29)(15,27)(16,30)(17,32,34,37)(19,26,20,36)(22,39,33,40)(24,35)(25,28)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fghi,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 32011);