#146228

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,d^2=g,e^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dgi,e^c=egi,f^c=fhi,g^c=gi

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, 53424);